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Chapter 8

Matrices and Determinants

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Problem 1

A rectangular array of real numbers that can be used to solve a system of linear equations is called
a ______.

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Problem 2

A matrix is _____ when the number of rows equals the number of columns.

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Problem 3

For a square matrix, the entries $a_{11}, a_{22}, a_{33}, \dots$ are the ________ ________ entries.

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Problem 4

A matrix with only one row is called a _______ matrix, and a matrix with only one column is called a ______ matrix.

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Problem 5

The matrix derived from a system of linear equations is called the _________ matrix of the system.

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Problem 6

The matrix derived from the coefficients of a system of linear equations is called the _______ matrix
of the system.

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Problem 7

. Two matrices are called _______ when one of the matrices can be obtained from the other by a
sequence of elementary row operations.

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Problem 8

. A matrix in row-echelon form is in _____ _____ ____ when every column that has a leading 1
has zeros in every position above and below its leading 1.

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Problem 9

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{ll}{7} & {0}\end{array}\right]$$

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Problem 10

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{llll}{5} & {-3} & {8} & {7}\end{array}\right]$$

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Problem 11

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{r}{2} \\ {36} \\ {3}\end{array}\right]$$

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Problem 12

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{rrrr}{-3} & {7} & {15} & {0} \\ {0} & {0} & {3} & {3} \\ {1} & {1} & {6} &{7}\end{array}\right]$$

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Problem 13

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{rr}{33} & {45} \\ {-9} & {20}\end{array}\right]$$

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Problem 14

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{rrr}{-7} & {6} & {4} \\ {0} & {-5} & {1}\end{array}\right]$$

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Problem 15

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{rrr}{1} & {6} & {-1} \\ {8} & {0} & {3} \\ {3} & {-9} & {9}\end{array}\right]$$

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Problem 16

Order of a Matrix, determine the order of the matrix.
$$\left[ \begin{array}{rr}{3} & {-1} \\ {4} & {1} \\ {-5} & {9}\end{array}\right]$$

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Problem 17

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{ccc}{4 x-3 y} & {=} & {-5} \\ {-x+3 y} & {=} & {12}\end{array}\right.$$

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Problem 18

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{l}{7 x+4 y=22} \\ {5 x-9 y=15}\end{array}\right.$$

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Problem 19

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{r}{x+10 y-2 z=2} \\ {5 x-3 y+4 z=0} \\ {2 x+y \quad=6}\end{array}\right.$$

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Problem 20

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{array}{rr}{-x-8 y+5 z=} & {8} \\ {-7 x-15 z=} & {-38} \\ {3 x-y+8 z=} & {20}\end{array}\right.$$

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Problem 21

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{aligned} 7 x-5 y+z &=13 \\ 19 x &-8 z=10 \end{aligned}\right.$$

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Problem 22

Writing an Augmented Matrix, write the augmented matrix for the system of linear equations.
$$\left\{\begin{aligned} 9 x+& 2 y-3 z=20 \\-25 y+11 z &=-5 \end{aligned}\right.$$

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Problem 23

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrr}{1} & {2} & {\vdots} & {7} \\ {2} & {-3} & {\vdots} & {4}\end{array}\right]$$

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Problem 24

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrr}{7} & {-5} & {\vdots} & {0} \\ {8} & {3} & {\vdots} & {-2}\end{array}\right]$$

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Problem 25

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrr}{2} & {0} & {5} & {\vdots} & {-12} \\ {0} & {1} & {-2} & {\vdots} & {7} \\ {6} & {3} & {0} & {\vdots} & {2}\end{array}\right]$$

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Problem 26

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrrr}{4} & {-5} & {-1} & {\vdots} & {18} \\ {-11} & {0} & {6} & {\vdots} & {25} \\ {3} & {8} & {0} & {\vdots} & {-29}\end{array}\right]$$

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Problem 27

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrrrr}{9} & {12} & {3} & {0} & {\vdots} & {0} \\ {-2} & {18} & {5} & {2} & {\vdots} & {10} \\ {1} & {7} & {-8} & {0} & {\vdots} & {-4} \\ {3} & {0} & {2} & {0} & {\vdots} & {-10}\end{array}\right]
$$

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Problem 28

Writing a System of Equations, write the system of linear equations represented by the augmented matrix. (Use variables $x, y, z,$ and $w,$ if applicable.)
$$\left[ \begin{array}{rrrrrr}{6} & {2} & {-1} & {-5} & {\vdots} & {-25} \\ {-1} & {0} & {7} & {3} & {\vdots} & {7} \\ {4} & {-1} & {-10} & {6} & {\vdots} & {23} \\ {0} & {8} & {1} & {-11} & {\vdots} & {-21}\end{array}\right]$$

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Problem 29

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$ New Row-Equivalent Matrix $\\$
$\left[ \begin{array}{rrr}{-2} & {5} & {1} \\ {3} & {-1} & {-8}\end{array}\right]$ $\quad$ $\quad$ $\quad$ $\left[ \begin{array}{rrr}{13} & {0} & {-39} \\ {3} & {-1} & {-8}\end{array}\right]$

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Problem 30

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$
$\left[ \begin{array}{rrr}{3} & {-1} & {-4} \\ {-4} & {3} & {7}\end{array}\right]$$\quad$$\quad$$\quad$$\quad$$\left[ \begin{array}{rrr}{3} & {-1} & {-4} \\ {5} & {0} & {-5}\end{array}\right]$

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Problem 31

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$ $\left[ \begin{array}{rrrr}{0} & {-1} & {-5} & {5} \\ {-1} & {3} & {-7} & {6} \\ {4} & {-5} & {1} & {3}\end{array}\right]$
$\quad$$\quad$$\quad$$\quad$$\quad$ $\left[ \begin{array}{rrrr}{-1} & {3} & {-7} & {6} \\ {0} & {-1} & {-5} & {5} \\ {0} & {7} & {-27} & {27}\end{array}\right]$

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Problem 32

Identifying an Elementary Row Operation, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix.
$\\$ Original Matrix $\quad$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$New Row-Equivalent Matrix $\\$$\left[ \begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {2} & {-5} & {1} & {-7} \\ {5} & {4} & {-7} & {6}\end{array}\right]$$\quad$$\quad$$\quad$$\quad$$\quad$$\left[ \begin{array}{rrrr}{-1} & {-2} & {3} & {-2} \\ {0} & {-9} & {7} & {-11} \\ {0} & {-6} & {8} & {-4}\end{array}\right]$

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Problem 33

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrr}{3} & {6} & {8} \\ {4} & {-3} & {6}\end{array}\right]$$
$$\left[ \begin{array}{rrr}{1} & {4} & {3} \\ {0} & {} & {-1}\end{array}\right]$$

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Problem 34

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrr}{3} & {6} & {8} \\ {4} & {-3} & {6}\end{array}\right]$$
$$\left[ \begin{array}{rrr}{1} & {} & {\frac{8}{3}} \\ {4} & {-3} & {{6}}\end{array}\right]$$

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Problem 35

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrr}{1} & {1} & {1} \\ {5} & {-2} & {4}\end{array}\right]$$
$$\left[ \begin{array}{rrr}{1} & {1} & {{1}} \\ {0} & {} & {-1}\end{array}\right]$$

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Problem 36

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrr}{-3} & {3} & {12} \\ {18} & {-8} & {4}\end{array}\right]$$
$$\left[ \begin{array}{rrr}{1} & {-1} & {} \\ {18} & {-8} & {4}\end{array}\right]$$

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Problem 37

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrrr}{1} & {5} & {4} & {-1} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} &{-7}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {0} & {} & {} \\ {0} & {1} & {-2} & {2} \\ {0} & {0} & {1} &{-7}\end{array}\right]$$

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Problem 38

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrrr}{1} & {0} & {6} & {1} \\ {0} & {-1} & {0} & {7} \\ {0} & {0} & {-1} &{3}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {0} & {6} & {1} \\ {0} & {1} & {0} & {} \\ {0} & {0} & {1} &{}\end{array}\right]$$

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Problem 39

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {3} & {8} & {10} & {3} \\ {-2} & {1} & {12} & {6}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {0} & {5} & {} & {} \\ {0} & {3} & {} & {}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {1} & {4} & {-1} \\ {0} & {1} & {-\frac{2}{5}} & {\frac{6}{5}} \\ {0} & {3} & {} & {}\end{array}\right]$$

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Problem 40

Elementary Row Operations, fill in the blank(s) using elementary row operations to form a
row-equivalent matrix.
$$\left[ \begin{array}{rrrr}{2} & {4} & {8} & {3} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {} & {} & {} \\ {1} & {-1} & {-3} & {2} \\ {2} & {6} & {4} & {9}\end{array}\right]$$
$$\left[ \begin{array}{rrrr}{1} & {2} & {4} & {\frac{3}{2}} \\ {0} & {} & {-7} & {\frac{1}{2}} \\ {0} & {2} & {} & {}\end{array}\right]$$

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Problem 41

Comparing Linear Systems and Matrix
Operations In Exercises 41 and $42,$ (a) perform the
row operations to solve the augmented matrix, (b) write
and solve the system of linear equations represented by
the augmented matrix, and (c) compare the two solution
methods. Which do you prefer?
$$\left[ \begin{array}{rrrr}{-3} & {4} & {\vdots} & {22} \\ {6} & {-4} & {\vdots} & {-28}\end{array}\right]$$
$$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\ {\text { (ii) Add }-2 \text { times } R_{1} \text { to } R_{2} \text { . }} \\ {\text { (iii) Multiply } R_{2} \text { by }-\frac{1}{4}} \\ {\text { (iv) Multiply } R_{1} \text { by } \frac{1}{3}}\end{array}$$

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Problem 42

Comparing Linear Systems and Matrix
Operations In Exercises 41 and $42,$ (a) perform the
row operations to solve the augmented matrix, (b) write
and solve the system of linear equations represented by
the augmented matrix, and (c) compare the two solution
methods. Which do you prefer?
$$\left[ \begin{array}{rrrr}{7} & {13} & {1} & {\vdots} & {-4} \\ {-3} & {-5} & {-1} & {\vdots} & {-4} \\ {3} & {6} & {1} & {\vdots} & {-2}\end{array}\right]$$
$$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\ {\text { (ii) Multiply } R_{1} \text { by } \frac{1}{4}} \\ {\text { (iii) } \text { Add } R_{3} \text { to } R_{2} \text { . }}\end{array}$$
$$\begin{array}{l}{\text { (iv) } \mathrm{Add}-3 \text { times } R_{1} \text { to } R_{3} \text { . }} \\ {\text { (v) } \mathrm{Add}-2 \text { times } R_{2} \text { to } R_{1} .}\end{array}$$

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Problem 43

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$\left[ \begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {5} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

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Problem 44

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$\left[ \begin{array}{llll}{1} & {3} & {0} & {0} \\ {0} & {0} & {1} & {8} \\ {0} & {0} & {0} & {0}\end{array}\right]$$

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Problem 45

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {-1} \\ {0} & {0} & {0} & {2}\end{array}\right]$$

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Problem 46

Row-Echelon Form, determine whether the matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form.
$$\left[ \begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {2} \\ {0} & {0} & {1} & {0}\end{array}\right]$$

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Problem 47

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[ \begin{array}{rrrr}{1} & {1} & {0} & {5} \\ {-2} & {-1} & {2} & {-10} \\ {3} & {6} & {7} & {14}\end{array}\right]$$

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Problem 48

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[ \begin{array}{rrrr}{1} & {2} & {-1} & {3} \\ {3} & {7} & {-5} & {14} \\ {-2} & {-1} & {-3} & {8}\end{array}\right]$$

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Problem 49

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[ \begin{array}{rrrr}{1} & {-1} & {-1} & {1} \\ {5} & {-4} & {1} & {8} \\ {-6} & {8} & {18} & {0}\end{array}\right]$$

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Problem 50

Writing a Matrix in Row-Echelon Form, write the matrix in row-echelon form. (Remember that the row-echelon form of a matrix is not unique.)
$$\left[ \begin{array}{rrrr}{1} & {-3} & {0} & {-7} \\ {-3} & {10} & {1} & {23} \\ {4} & {-10} & {2} & {-24}\end{array}\right]$$

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Problem 51

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{rrr}{3} & {3} & {3} \\ {-1} & {0} & {-4} \\ {2} & {4} & {-2}\end{array}\right]$$

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Problem 52

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{ccc}{1} & {3} & {2} \\ {5} & {15} & {9} \\ {2} & {6} & {10}\end{array}\right]$$

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Problem 53

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{rrrr}{1} & {2} & {3} & {-5} \\ {1} & {2} & {4} & {-9} \\ {-2} & {-4} & {-4} & {3} \\ {4} & {8} & {11} & {-14}\end{array}\right]$$

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Problem 54

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{rrrr}{-2} & {3} & {-1} & {-2} \\ {4} & {-2} & {5} & {8} \\ {1} & {5} & {-2} & {0} \\ {3} & {8} & {-10} & {-30}\end{array}\right]$$

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Problem 55

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{rrrr}{-3} & {5} & {1} & {12} \\ {1} & {-1} & {1} & {4}\end{array}\right]$$

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Problem 56

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.
$$\left[ \begin{array}{rrr}{5} & {1} & {2} & {4} \\ {-1} & {5} & {10} & {-32}\end{array}\right]$$

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Problem 57

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)
$$\left[ \begin{array}{rrrr}{1} & {-2} & {\vdots} & {4} \\ {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

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Problem 58

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)
$$\left[ \begin{array}{llll}{1} & {5} & {\vdots} & {0} \\ {0} & {1} & {\vdots} & {-1}\end{array}\right]$$

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Problem 59

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)
$$\left[ \begin{array}{rrrrr}{1} & {-1} & {2} & {\vdots} & {4} \\ {0} & {1} & {-1} & {\vdots} & {2} \\ {0} & {0} & {1} & {\vdots} & {-2}\end{array}\right]$$

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Problem 60

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x, y,$ and $z,$ if applicable.)
$$\left[ \begin{array}{rrrrr}{1} & {2} & {-2} & {\vdots} & {-1} \\ {0} & {1} & {1} & {\vdots} & {9} \\ {0} & {0} & {1} & {\vdots} & {-3}\end{array}\right]$$

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Problem 61

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan
elimination. Write the solution represented by the augmented matrix.
$$\left\{\begin{aligned} 3 x+3 y+12 z &=6 \\ x+y+4 z &=2 \\ 2 x+5 y+20 z &=10 \\-x+2 y+8 z &=4 \end{aligned}\right.$$

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Problem 62

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan
elimination. Write the solution represented by the augmented matrix.
$$\left[ \begin{array}{cccc}{1} & {0} & {\vdots} & {-6} \\ {0} & {1} & {\vdots} & {10}\end{array}\right]$$

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Problem 63

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan
elimination. Write the solution represented by the augmented matrix.
$$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {\vdots} & {-4} \\ {0} & {1} & {0} & {\vdots} & {-10} \\ {0} & {0} & {1} & {\vdots} & {4}\end{array}\right]$$

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Problem 64

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables $x, y,$ and $z,$ if applicable) has been reduced using Gauss-Jordan
elimination. Write the solution represented by the augmented matrix.
$$\left[ \begin{array}{rrrrr}{1} & {0} & {0} & {\vdots} & {5} \\ {0} & {1} & {0} & {\vdots} & {-3} \\ {0} & {0} & {1} & {\vdots} & {0}\end{array}\right]$$

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Problem 65

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.$$

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Problem 66

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{array}{r}{2 x+6 y=16} \\ {2 x+3 y=7}\end{array}\right.$$

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Problem 67

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{array}{r}{3 x-2 y=-27} \\ {x+3 y=13}\end{array}\right.$$

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Problem 68

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned}-x+y &=4 \\ 2 x-4 y &=-34 \end{aligned}\right.$$

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Problem 69

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{array}{rr}{x+2 y-3 z=} & {-28} \\ {4 y+2 z=} & {0} \\ {-x+y-z=} & {-5}\end{array}\right.$$

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Problem 70

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned} 3 x-2 y+z=& 15 \\-x+y+2 z=&-10 \\ x-y-4 z=& 14 \end{aligned}\right.$$

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Problem 71

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{array}{rr}{-x+y=} & {-22} \\ {3 x+4 y=} & {4} \\ {4 x-8 y=} & {32}\end{array}\right.$$

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Problem 72

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.$$

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Problem 73

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned} 3 x+2 y-z+w &=0 \\ x-y+4 z+2 w &=25 \\-2 x+y+2 z-w &=2 \\ x+y+z+w &=6 \end{aligned}\right.$$

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Problem 74

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution.
$$\left\{\begin{aligned} x-4 y+3 z-2 w=& 9 \\ 3 x-2 y+z-4 w=&-13 \\-4 x+3 y-2 z+w=&-4 \\-2 x+y-4 z+3 w=&-10 \end{aligned}\right.$$

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Problem 75

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned}-2 x+6 y &=-22 \\ x+2 y &=-9 \end{aligned}\right.$$

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Problem 76

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned} 5 x-5 y &=-5 \\-2 x-3 y &=7 \end{aligned}\right.$$

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Problem 77

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{array}{l}{8 x-4 y=7} \\ {5 x+2 y=1}\end{array}\right.$$

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Problem 78

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{array}{rr}{x-3 y=} & {5} \\ {-2 x+6 y=} & {-10}\end{array}\right.$$

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Problem 79

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{array}{c}{x+2 y+z=8} \\ {3 x+7 y+6 z=26}\end{array}\right.$$

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Problem 80

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned} x+y+4 z &=5 \\ 2 x+y-z &=9 \end{aligned}\right.$$

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Problem 81

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned} x &-3 z=-2 \\ 3 x+y-2 z &=5 \\ 2 x+2 y+z &=4 \end{aligned}\right.$$

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Problem 82

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned} 2 x-y+3 z &=24 \\ 2 y-z &=14 \\ 7 x-5 y &=6 \end{aligned}\right.$$

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Problem 83

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{array}{rr}{-x+y-z=} & {-14} \\ {2 x-y+z=} & {21} \\ {3 x+2 y+z=} & {19}\end{array}\right.$$

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Problem 84

Gauss-Jordan Elimination, use matrices to solve the system of equations (if possible). Use Gauss-Jordan elimination.
$$\left\{\begin{aligned} 2 x+2 y-z &=2 \\ x-3 y+z &=-28 \\-x+y &=14 \end{aligned}\right.$$

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Problem 85

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{aligned} 3 x+3 y+12 z &=6 \\ x+y+4 z &=2 \\ 2 x+5 y+20 z &=10 \\-x+2 y+8 z &=4 \end{aligned}\right.$$

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Problem 86

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{aligned} 2 x+10 y+2 z=& 6 \\ x+5 y+2 z=& 6 \\ x+5 y+z=& 3 \\-3 x-15 y-3 z=&-9 \end{aligned}\right.$$

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Problem 87

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{aligned} 2 x+y-z+2 w &=-6 \\ 3 x+4 y &+w=& 1 \\ x+5 y+2 z+6 w &=-3 \\ 5 x+2 y-z-w &=3 \end{aligned}\right.$$

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Problem 88

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{aligned} x+2 y+2 z+4 w=& 11 \\ 3 x+6 y+5 z+12 w=& 30 \\ x+3 y-3 z+2 w=&-5 \\ 6 x-y-z+\quad w=&-9 \end{aligned}\right.$$

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Problem 89

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{aligned} x+y+z+w &=0 \\ 2 x+3 y+z-2 w &=0 \\ 3 x+5 y+z &=0 \end{aligned}\right.$$

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Problem 90

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system.
$$\left\{\begin{array}{r}{x+2 y+z+3 w=0} \\ {x-y+w=0} \\ {y-z+2 w=0}\end{array}\right.$$

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Problem 91

Comparing Solutions of Two Systems, determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{rr}{x-2 y+z=} & {-6} \\ {y-5 z=} & {16} \\ {z=} & {-3}\end{array}\right.\quad(b)
\left\{\begin{aligned} x+y-2 z &=6 \\ y+3 z &=-8 \\ z &=-3 \end{aligned}\right.$$

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Problem 92

Comparing Solutions of Two Systems In, determine whether the two systems of
linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{rr}{x-3 y+4 z=} & {-11} \\ {y-z=} & {-4} \\ {z=} & {2}\end{array}\right.$$
$$(b)\left\{\begin{array}{rr}{x+4 y} & {=-11} \\ {y+3 z} & {=4} \\ {z} & {=2}\end{array}\right.$$

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Problem 93

Comparing Solutions of Two Systems In, determine whether the two systems of
linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{rr}{x-4 y+5 z=} & {27} \\ {y-7 z=} & {-54} \\ {z=} & {8}\end{array}\right.$$
$$(b)\left\{\begin{aligned} x-6 y+z &=15 \\ y+5 z &=42 \\ z &=8 \end{aligned}\right.$$

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Problem 94

Comparing Solutions of Two Systems In, determine whether the two systems of
linear equations yield the same solution. If so, find the solution using matrices.
$$(a)\left\{\begin{array}{rrr}{x+3 y-z=} & {19} \\ {y+6 z=} & {-18} \\ {z=} & {-4}\end{array}\right.$$
$$(b)\left\{\begin{array}{rrr}{x-y+3 z=} & {-15} \\ {y-2 z=} & {14} \\ {z=} & {-4}\end{array}\right.$$

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Problem 95

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(1)=1, f(2)=-1, f(3)=-5$$

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Problem 96

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(1)=2, f(2)=9, f(3)=20$$

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Problem 97

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(-2)=-15, f(-1)=7, f(1)=-3$$

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Problem 98

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(-2)=-3, f(1)=-3, f(2)=-11$$

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Problem 99

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(1)=8, f(2)=13, f(3)=20$$

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Problem 100

Curve Fitting, use a system of equations to find the quadratic function $f(x)=a x^{2}+b x+c$ that satisfies the given conditions. Solve the system using matrices.
$$f(1)=9, f(2)=8, f(3)=5$$

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Problem 101

From 2000 through $2011,$ the numbers of new cases
of a waterborne disease in a small city increased in
a pattern that was approximately linear (see figure).
Find the least squares regression line
$$y=a t+b$$
for the data shown in the figure by solving the following system using matrices. Let $t$ represent the year, with $t=0$ corresponding to $2000 .$
$$\left\{\begin{aligned} 12 b+66 a &=831 \\ 66 b+506 a &=5643 \end{aligned}\right.$$
Use the result to predict the number of new cases
of the waterbome disease in $2014 .$ Is the estimate
reasonable? Explain.

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Problem 102

Breeding Facility A city zoo borrowed
$\$ 2,000,000$ at simple annual interest to construct a
breeding facility. Some of the money was borrowed at
$8 \%,$ some at $9 \%,$ and some at 12$\% .$ Use a system of
linear equations to determine how much was borrowed at each rate given that the total annual interest was
$\$ 186,000$ and the amount borrowed at 8$\%$ was twice
the amount borrowed at 12$\% .$ Solve the system of
linear equations using matrices.

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Problem 103

Museum A natural history museum borrowed
$\$ 2,000,000$ at simple annual interest to purchase new
exhibits. Some of the money was borrowed at $7 \%,$
some at $8.5 \%,$ and some at 9.5$\% .$ Use a system of
linear equations to determine how much was borrowed
at each rate given that the total annual interest was $\$ 169,750$ and the amount borrowed at 8.5$\%$ was four
times the amount borrowed at 9.5$\% .$ Solve the system
of linear equations using matrices.

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Problem 104

Mathematical Modeling A video of the path of
a ball thrown by a baseball player was analyzed with a
grid covering the TV screen. The tape was paused
three times, and the position of the ball was measured
each time. The coordinates obtained are shown in the
table. $(x$ and $y$ are measured in feet.)
$$\begin{array}{|c|c|c|c|c|}\hline \text { Horizontal Distance, } x & {0} & {15} & {30} \\ \hline \text { Height, y } & {5.0} & {9.6} & {12.4} \\ \hline\end{array}$$
$$\begin{array}{l}{\text { (a) Use a system of equations to find the equation of the }} \\ {\text { parabola } y=a x^{2}+b x+c \text { that passes through the }} \\ {\text { three points. Solve the system using matrices. }} \\ {\text { (b) Use a graphing utility to graph the parabola. }}\end{array}$$
$$\begin{array}{l}{\text { (c) Graphically approximate the maximum height of the }} \\ {\text { ball and the point at which the ball struck the ground. }} \\ {\text { (d) Analytically find the maximum height of the ball }} \\ {\text { and the point at which the ball struck the ground. }} \\ {\text { (e) Compare your results from parts (c) and (d). }}\end{array}$$

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Problem 105

True or False?, determine whether the statement is true or false. Justify your answer.
$\left[ \begin{array}{cccc}{5} & {0} & {-2} & {7} \\ {-1} & {3} & {-6} & {0}\end{array}\right]$ is a $4 \times 2$ matrix.

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Problem 106

The method of Gaussian elimination reduces a matrix
until a reduced row-echelon form is obtained.

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Problem 107

Think About It What is the relationship between
the three elementary row operations performed on an
augmented matrix and the operations that lead to
equivalent systems of equations?

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Problem 108

HOW DO YOU SEEIT? Determine whether
the matrix below is in row-echelon form,
reduced row-echelon form, or neither when
it satisfies the given conditions.
$$\left[ \begin{array}{ll}{1} & {b} \\ {c} & {1}\end{array}\right]$$
$$\begin{array}{ll}{\text { (a) } b=0, c=0} & {\text { (b) } b \neq 0, c=0} \\ {\text { (c) } b=0, c \neq 0} & {\text { (d) } b \neq 0, c \neq 0}\end{array}$$

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