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Pre-Calculus

StemEZ

Chapter 25

Determinates, Matrices And System Of Equations - all with Video Answers

Educators

Chapter Questions

00:48

Problem 587

Solve the following linear equations by using Cramer's Rule:
$-2 \mathrm{x}^{1}+3 \mathrm{x}_{2}-\mathrm{x}_{3}=1$
$\mathrm{x}_{1}+2 \mathrm{x}_{2}-\mathrm{x}_{3}=4$
$-2 \mathrm{x}_{1}-\mathrm{x}_{2}+\mathrm{x}_{3}=-3$

James Kiss
James Kiss
Numerade Educator
00:55

Problem 588

Solve the following homogeneous equations:
$$
\begin{array}{r}
\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}=0 \\
\mathrm{x}_{2}-3 \mathrm{x}_{3}=0 \\
-\mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}=0
\end{array}
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:49

Problem 589

Solve the system of linear equations:
$3 x+2 y+4 z=1$
$2 \mathrm{x}-\mathrm{y}+\mathrm{z}=0$
$x+2 y+3 z=1$

David Mccaslin
David Mccaslin
Numerade Educator
00:55

Problem 590

Let the homogeneous linear $A X=B$ be given by
$\begin{array}{lll}\mid 1 & 2 & 0 \mid & \left|x_{1}\right| \quad|0| \\ \mid 0 & 1 & 3 \mid & \left|x_{2}\right|=|0| \\ \mid 2 & 1 & 3 \mid & \left|x_{3}\right| \quad|0|\end{array}$
Show that $A$ has only the trivial solution, $\left(x_{1}, x_{2}, x_{3}\right)=(0,0,0)$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:34

Problem 591

Show that the system has a solution without actually computing a solution.

Sikandar Baig
Sikandar Baig
Numerade Educator
02:04

Problem 592

Solve the following system of equations by forming the matrix of coefficients and reducing it to echelon form. $3 x+2 y-z=0$
$\mathrm{x}-\mathrm{y}+2 \mathrm{z}=0$
$x+y-6 z=0$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:23

Problem 593

Let $A=\left|\begin{array}{ccc}1 & 2 & -1 \\ \mid-2 & -1 & 1\end{array}\right|$ be the coefficient matrix of a homogeneous system in $\mathrm{x}, \mathrm{y}$, and $\mathrm{z}$. Solve this system to illustrate that a homogeneous system of 3 equations in the unknowns, $\mathrm{x}, \mathrm{y}, \mathrm{z}$ has a unique solution.

Vysakh M
Vysakh M
Numerade Educator
03:19

Problem 594

Solve the following system of equations:
$x+3 y=0$
$2 x+6 y+4 z=0$
(1)

AG
Ankit Gupta
Numerade Educator
01:07

Problem 595

Solve the following homogeneous system of linear equations.
$$
\begin{gathered}
2 \mathrm{x}_{1}+2 \mathrm{x}_{2}-\mathrm{x}_{3}+\mathrm{x}_{5}=0 \\
-\mathrm{x}_{1}+\mathrm{x}_{2}+2 \mathrm{x}_{3}-3 \mathrm{x}_{4}+\mathrm{x}_{5}=0 \\
\mathrm{x}_{1}+\mathrm{x}_{3}-2 \mathrm{x}_{3}-\mathrm{x}_{5}=0 \\
\mathrm{x}_{3}+\mathrm{x}_{4}+\mathrm{x}_{5}=0
\end{gathered}
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
10:54

Problem 596

[A] Show that each of the following systems has a nonzero solution:
(a) $\mathrm{x}+\mathrm{y}-3 \mathrm{z}+\mathrm{w}=0$
$\mathrm{x}-\mathrm{y}+\mathrm{z}-\mathrm{w}=0$
$2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}+\mathrm{w}=0$
(b) $x+y-z=0$
$2 x-3 y+z=0$
$x-4 y+2 z=0$
[B] Show that following system has a unique solution:
$\mathrm{x}+\mathrm{y}-\mathrm{z}=0$
$2 \mathrm{x}+4 \mathrm{y}-\mathrm{z}=0$
$3 \mathrm{x}+2 \mathrm{v}+2 \mathrm{z}=0$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 597

By forming the augmented matrix and row reducing, determine the solutions of the following system $2 x-y+3 z=4$
$3 \mathrm{x}+2 z=5$
$-2 x+y+4 z=6$

Raj Bala
Raj Bala
Numerade Educator
03:33

Problem 598

Show that the following non-homogeneous system of linear equations has no solution $x+2 y-3 z=-1$
$3 x-y+2 z=7$
$5 x+3 y-4 z=2$

AG
Ankit Gupta
Numerade Educator
03:04

Problem 599

Solve the following system by Gauss-Jordan elimination
$$
\begin{aligned}
x_{1}+3 x_{2}-2 x_{3}+2 x_{5} &=0 \\
2 x_{1}+6 x_{2}-5 x_{3}-2 x_{4}+5 x_{5}-6 x_{6} &=-1 \\
5 x_{3}+10 x_{4}+15 x_{6} &=5 \\
2 x_{1}+6 x_{2}+8 x_{4}+4 x_{5}+18 x_{6} &=6
\end{aligned}
$$

James Kiss
James Kiss
Numerade Educator
01:09

Problem 600

Show that the following system has more than one solution. $3 x-y+7 z=0$
$2 x-y+4 z=1 / 2$
$\mathrm{x}-\mathrm{y}+\mathrm{z}=1$
$6 x-4 y+10 z=3$

James Kiss
James Kiss
Numerade Educator
06:45

Problem 601

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given
(b) $\begin{array}{ll}10 & 1 \\ 1 & 3|3|\end{array}$
(c) $\left|\begin{array}{rrrrr|r}1 & 6 & 0 & 0 & 4 & -2 \mid \\ 0 & 0 & 1 & 0 & 3 & 1 \\ 0 & 0 & 0 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right|$
(d) $\left|\begin{array}{llll}0 & \mid 1 & 0 & 0|0| \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{array}\right|^{2}$

Jillian Rae Villa
Jillian Rae Villa
Numerade Educator
02:33

Problem 602

Find the necessary and sufficient conditions for the existence of a solution to the following system. $\mathrm{x}+\mathrm{y}+2 \mathrm{z}=\mathrm{a}_{1}$
$-2 \mathrm{x} \quad-\mathrm{z}=\mathrm{a}_{2}$
$\mathrm{x}+3 \mathrm{y}+5 \mathrm{z}=\mathrm{a}_{3}$

Jenna Dula
Jenna Dula
Numerade Educator
04:40

Problem 603

Determine the values of a so that the following system of equations has: (a) no solution, (b) more than one solution,
(c) a unique solution. $x+y-z=1$
$2 x+3 y+a z=3$
$x+a y+3 z=2$

Jenna Dula
Jenna Dula
Numerade Educator
02:22

Problem 604

Solve the following system $\mathrm{x}_{1}-2 \mathrm{x}_{2}-3 \mathrm{x}_{3}=3$
$2 \mathrm{x}_{1}-\mathrm{x}_{2}-4 \mathrm{x}_{3}=7$
$3 \mathrm{x}_{1}-3 \mathrm{x}_{2}-5 \mathrm{x}_{3}=8$

Ashley Volpe
Ashley Volpe
Numerade Educator
10:28

Problem 605

Consider the following nonhomogeneous system of linear equations. $2 x+y-3 z=1$
$3 \mathrm{x}+2 \mathrm{y}-2 \mathrm{z}=2$
$x+y+z=1$
Show that (i) any two solutions to the system (1) differ by a vector which is a solution to the homogeneous system $2 x+y-3 z=0$
$3 \mathrm{x}+2 \mathrm{y}-2 \mathrm{z}=0$
$x+y+z=0$
(ii) the sum of a solution to (1) and a solution to
(2) gives a solution to (1).

Anthony Ramos
Anthony Ramos
Numerade Educator
05:14

Problem 606

If the method of Gauss elimination corresponds in its final form to an echelon matrix, what is the matrix analogue of the Gauss-Jordan method for solving linear systems of equations? Explain by example.

Jennifer Stoner
Jennifer Stoner
Numerade Educator