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Elementary Linear Algebra

Ron Larson

Chapter 3

Determinants - all with Video Answers

Educators

Section 1

The Determinant of a Matrix

00:55

Problem 1

Find the determinant of the matrix.
$$[1]$$

Anthony Ramos
Anthony Ramos
Numerade Educator
00:49

Problem 2

Find the determinant of the matrix.
$$[-3]$$

Thomas Emment
Thomas Emment
Numerade Educator
04:47

Problem 3

Find the determinant of the matrix.
$$\left[\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
05:09

Problem 4

Find the determinant of the matrix.
$$\left[\begin{array}{rr}-3 & 1 \\ 5 & 2\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
00:31

Problem 5

Find the determinant of the matrix.
$$\left[\begin{array}{rr}5 & 2 \\ -6 & 3\end{array}\right]$$

AG
Ankit Gupta
Numerade Educator
05:22

Problem 6

Find the determinant of the matrix.
$$\left[\begin{array}{rr}2 & -2 \\ 4 & 3\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
00:39

Problem 7

Find the determinant of the matrix.
$$\left[\begin{array}{rr}-7 & 6 \\ \frac{1}{2} & 3\end{array}\right]$$

AG
Ankit Gupta
Numerade Educator
04:42

Problem 8

Find the determinant of the matrix.
$$\left[\begin{array}{rr}\frac{1}{3} & 5 \\ 4 & -9\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
05:08

Problem 9

Find the determinant of the matrix.
$$\left[\begin{array}{ll}0 & 8 \\ 0 & 4\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
01:23

Problem 10

Find the determinant of the matrix.
$$\left[\begin{array}{rr}2 & -3 \\ -6 & 9\end{array}\right]$$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:49

Problem 11

Find the determinant of the matrix.
$$\left[\begin{array}{rrr}\lambda-3 & 2 \\ 4 & \lambda-1\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
04:29

Problem 12

Find the determinant of the matrix.
$$\left[\begin{array}{rr}\lambda-2 & 0 \\ 4 & \lambda-4\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
03:07

Problem 13

Find all (a) minors and (b) cofactors of the matrix.
$$\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:35

Problem 14

Find all (a) minors and (b) cofactors of the matrix.
$$\left[\begin{array}{rr}-5 & 6 \\ 1 & 0\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
06:48

Problem 15

Find all (a) minors and (b) cofactors of the matrix.
$$\left[\begin{array}{rrr}-3 & 2 & 1 \\ 4 & 5 & 6 \\ 2 & -3 & 1\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
05:05

Problem 16

Find all (a) minors and (b) cofactors of the matrix.
$$\left[\begin{array}{rrr}-3 & 4 & 2 \\ 6 & 3 & 1 \\ 4 & -7 & -8\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
04:23

Problem 17

Find the determinant of the matrix in Exercise 15 using the method of expansion by cofactors. Use (a) the second row and (b) the second column.

Syed Basim Mehmood
Syed Basim Mehmood
Numerade Educator
03:22

Problem 18

Find the determinant of the matrix in Exercise 16 using the method of expansion by cofactors. Use (a) the third row and (b) the first column.

Syed Basim Mehmood
Syed Basim Mehmood
Numerade Educator
03:44

Problem 19

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
1 & 4 & -2 \\
3 & 2 & 0 \\
-1 & 4 & 3
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:22

Problem 20

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
3 & -1 & 2 \\
4 & 1 & 4 \\
-2 & 0 & 1
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:12

Problem 21

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
2 & 4 & 6 \\
0 & 3 & 1 \\
0 & 0 & -5
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:41

Problem 22

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
-3 & 0 & 0 \\
7 & 11 & 0 \\
1 & 2 & 2
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:45

Problem 23

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
-0.4 & 0.4 & 0.3 \\
0.2 & 0.2 & 0.2 \\
0.3 & 0.2 & 0.2
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
04:45

Problem 24

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
0.1 & 0.2 & 0.3 \\
-0.3 & 0.2 & 0.2 \\
0.5 & 0.4 & 0.4
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:28

Problem 25

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
x & y & -1 \\
3 & 2 & 0 \\
1 & 1 & 1
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:14

Problem 26

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
x & y & 1 \\
-2 & -2 & 1 \\
1 & 5 & 1
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
05:06

Problem 27

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrrr}
5 & 3 & 0 & 6 \\
4 & 6 & 4 & 12 \\
0 & 2 & -3 & 4 \\
0 & 1 & -2 & 2
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:24

Problem 28

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrrr}
3 & 0 & 7 & 0 \\
2 & 6 & 11 & 12 \\
4 & 1 & -1 & 2 \\
1 & 5 & 2 & 10
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
09:24

Problem 29

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrrr}
w & x & y & z \\
21 & -15 & 24 & 30 \\
-10 & 24 & -32 & 18 \\
-40 & 22 & 32 & -35
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
07:36

Problem 30

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrrr}
w & x & y & z \\
10 & 15 & -25 & 30 \\
-30 & 20 & -15 & -10 \\
30 & 35 & -25 & -40
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
05:06

Problem 31

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{ccccc}
5 & 2 & 0 & 0 & -2 \\
0 & 1 & 4 & 3 & 2 \\
0 & 0 & 2 & 6 & 3 \\
0 & 0 & 3 & 4 & 1 \\
0 & 0 & 0 & 0 & 2
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
05:06

Problem 32

Use expansion by cofactors to find the determinant of the matrix.
$$\left[\begin{array}{rrrrr}
-4 & 3 & 2 & -1 & -2 \\
1 & -2 & 7 & -13 & -12 \\
-6 & 2 & -5 & -6 & -7 \\
0 & 0 & 0 & 0 & 0 \\
1 & -4 & -2 & 0 & -9
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:21

Problem 33

Use the method demonstrated in Example 5 to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
3 & 0 & 4 \\
-2 & 4 & 1 \\
1 & -3 & 1
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
04:49

Problem 34

Use the method demonstrated in Example 5 to find the determinant of the matrix.
$$\left[\begin{array}{rrr}
3 & 8 & -7 \\
0 & -5 & 4 \\
8 & 1 & 6
\end{array}\right]$$

Sanchit Jain
Sanchit Jain
Numerade Educator
01:55

Problem 35

Use a software program or a graphing utility to find the matrix.
$$\left[\begin{array}{rrr}
0.1 & 0.6 & -0.3 \\
0.7 & -0.1 & 0.1 \\
0.1 & 0.3 & -0.8
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
05:50

Problem 36

Use a software program or a graphing utility to find the matrix.
$$\left[\begin{array}{rrrr}
4 & 3 & 2 & 5 \\
1 & 6 & -1 & 2 \\
-3 & 2 & 4 & 5 \\
6 & 1 & 3 & -2
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
04:08

Problem 37

Use a software program or a graphing utility to find the matrix.
$$\left|\begin{array}{rrrr}
1 & 2 & -1 & 4 \\
0 & 1 & 2 & -2 \\
0 & 3 & 2 & -1 \\
1 & 2 & 0 & -2
\end{array}\right|$$

Chandra Jain
Chandra Jain
Numerade Educator
05:50

Problem 38

Use a software program or a graphing utility to find the matrix.
$$\left[\begin{array}{rrrrr}
0 & 5 & 1 & -2 & 0 \\
-1 & 0 & 7 & 1 & 6 \\
0 & 8 & 6 & 5 & -3 \\
1 & 2 & 5 & -8 & 4 \\
2 & 6 & -2 & 0 & 6
\end{array}\right]$$

Chandra Jain
Chandra Jain
Numerade Educator
02:49

Problem 39

Find the determinant of the triangular matrix.
$$\left[\begin{array}{rrr}
-2 & 0 & 0 \\
4 & 6 & 0 \\
-3 & 7 & 2
\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
03:12

Problem 40

Find the determinant of the triangular matrix.
$$\left[\begin{array}{rrr}
4 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & -2
\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
03:05

Problem 41

Find the determinant of the triangular matrix.
$$\left[\begin{array}{rrrr}
5 & 8 & -4 & 2 \\
0 & 0 & 6 & 0 \\
0 & 0 & 2 & 2 \\
0 & 0 & 0 & -1
\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
03:50

Problem 42

Find the determinant of the triangular matrix.
$$\left[\begin{array}{rrrr}
4 & 1 & 0 & 0 \\
-1 & \frac{1}{2} & 0 & 0 \\
3 & 5 & 3 & 0 \\
-8 & 7 & 0 & -2
\end{array}\right]$$

Asif Khan
Asif Khan
Numerade Educator
05:05

Problem 43

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
$\begin{array}{lllllll}\text { (a) The } & \text { determinant } & \text { of } & \text { a } & 2 \times 2 & \text { matrix } & A & \text { is }\end{array}$
$$a_{21} a_{12}-a_{11} a_{22}$$
(b) The determinant of a matrix of order 1 is the entry of the matrix.
(c) The ij-cofactor of a square matrix $A$ is the matrix obtained by deleting the ith row and jth column of $A$.

Sanchit Jain
Sanchit Jain
Numerade Educator
01:21

Problem 44

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) To find the determinant of a triangular matrix, add the entries on the main diagonal.
(b) To find the determinant of a matrix, expand by cofactors in any row or column.
(c) When expanding by cofactors, you need not evaluate the cofactors of zero entries

Chandra Jain
Chandra Jain
Numerade Educator
02:27

Problem 45

Solve for $x .$
$$\left|\begin{array}{rr}
x+3 & 2 \\
1 & x+2
\end{array}\right|=0$$

AG
Ankit Gupta
Numerade Educator
04:49

Problem 46

Solve for $x .$
$$\left|\begin{array}{rr}
x-6 & 3 \\
-2 & x+1
\end{array}\right|=0$$

Asif Khan
Asif Khan
Numerade Educator
01:07

Problem 47

Solve for $x .$
$$\left|\begin{array}{rr}
x-1 & 2 \\
3 & x-2
\end{array}\right|=0$$

AG
Ankit Gupta
Numerade Educator
05:57

Problem 48

Solve for $x .$
$$\left|\begin{array}{rr}
x+3 & 1 \\
-4 & x-1
\end{array}\right|=0$$

Asif Khan
Asif Khan
Numerade Educator
06:47

Problem 49

Find the values of $\lambda$ for which the determinant is zero.
$$\left|\begin{array}{cc}
\lambda+2 & 2 \\
1 & \lambda
\end{array}\right|$$

Asif Khan
Asif Khan
Numerade Educator
07:55

Problem 50

Find the values of $\lambda$ for which the determinant is zero.
$$\left|\begin{array}{rr}
\lambda-5 & 3 \\
1 & \lambda-5
\end{array}\right|$$

Asif Khan
Asif Khan
Numerade Educator
02:59

Problem 51

Find the values of $\lambda$ for which the determinant is zero.
$$\left|\begin{array}{lll}
\lambda & 2 & 0 \\
0 & \lambda+1 & 2 \\
0 & 1 & \lambda
\end{array}\right|$$

Sanchit Jain
Sanchit Jain
Numerade Educator
03:11

Problem 52

Find the values of $\lambda$ for which the determinant is zero.
$$\left|\begin{array}{rrr}
\lambda & 0 & 1 \\
0 & \lambda & 3 \\
2 & 2 & \lambda-2
\end{array}\right|$$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:16

Problem 53

Show that the system of linear equations
$a_{11} x_{1}+a_{12} x_{2}=b_{1}$
$a_{21} x_{1}+a_{22} x_{2}=b_{2}$
has the solution
$x_{1}=\frac{b_{1} a_{22}-b_{2} a_{12}}{a_{11} a_{22}-a_{21} a_{12}}$ and $x_{2}=\frac{b_{2} a_{11}-b_{1} a_{21}}{a_{11} a_{22}-a_{21} a_{12}}$
when $a_{11} a_{22}-a_{21} a_{12} \neq 0$

Aayush Gupta
Aayush Gupta
Numerade Educator
02:18

Problem 54

For an $n \times n$ matrix $A$, explain how to find each value.
(a) The minor $M_{i j}$ of the entry $a_{i j}$
(b) The cofactor $C_{i j}$ of the entry $a_{i j}$
(c) The determinant of $A$

Aayush Gupta
Aayush Gupta
Numerade Educator
01:07

Problem 55

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{cc}
6 u & -1 \\
-1 & 3 v
\end{array}\right|$$

Sanchit Jain
Sanchit Jain
Numerade Educator
01:01

Problem 56

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{cc}
3 x^{2} & -3 y^{2} \\
1 & 1
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
01:23

Problem 57

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{cc}
e^{2 x} & e^{3 x} \\
2 e^{2 x} & 3 e^{3 x}
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
01:28

Problem 58

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{cc}
e^{-x} & x e^{-x} \\
-e^{-x} & (1-x) e^{-x}
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
00:32

Problem 59

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{ll}
x & \ln x \\
1 & 1 / x
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 60

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{rr}
x & x \ln x \\
1 & 1+\ln x
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
02:00

Problem 61

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{ccc}
\cos \theta & -r \sin \theta & 0 \\
\sin \theta & r \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right|$$

Sanchit Jain
Sanchit Jain
Numerade Educator
04:51

Problem 62

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.
$$\left|\begin{array}{ccc}
1-v & -u & 0 \\
v(1-w) & u(1-w) & -u v \\
v w & u w & u v
\end{array}\right|$$

Sanchit Jain
Sanchit Jain
Numerade Educator
01:22

Problem 63

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{ll}
w & x \\
y & z
\end{array}\right|=-\left|\begin{array}{ll}
y & z \\
w & x
\end{array}\right|$$

Anas Venkitta
Anas Venkitta
Numerade Educator
01:57

Problem 64

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{ll}
w & c x \\
y & c z
\end{array}\right|=c\left|\begin{array}{ll}
w & x \\
y & z
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
02:04

Problem 65

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{ll}
w & x \\
y & z
\end{array}\right|=\left|\begin{array}{ll}
w & x+c w \\
y & z+c y
\end{array}\right|$$

AG
Ankit Gupta
Numerade Educator
00:56

Problem 66

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{ll}
w & x \\
c w & c x
\end{array}\right|=0$$

AG
Ankit Gupta
Numerade Educator
02:49

Problem 67

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{lll}
1 & x & x^{2} \\
1 & y & y^{2} \\
1 & z & z^{2}
\end{array}\right|=(y-x)(z-x)(z-y)$$

AG
Ankit Gupta
Numerade Educator
05:53

Problem 68

Evaluate the determinants to verify the equation.
$$\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^{3} & b^{3} & c^{3}
\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$$

Sanchit Jain
Sanchit Jain
Numerade Educator
05:00

Problem 69

You are given the equation
$$\left|\begin{array}{rrr}
x & 0 & c \\
-1 & x & b \\
0 & -1 & a
\end{array}\right|=a x^{2}+b x+c.$$
(a) Verify the equation.
(b) Use the equation as a model to find a determinant that is equal to $a x^{3}+b x^{2}+c x+d$

Sanchit Jain
Sanchit Jain
Numerade Educator
02:18

Problem 70

The determinant of a $2 \times 2$ matrix involves two Products. The determinant of a $3 \times 3$ matrix involves six triple products. Show that the determinant of a $\exists \times 4$ matrix involves 24 quadruple products.

M S
M S
Numerade Educator