In each part of Exercises 11-12, find matrices A, x, and b that express the given linear system as a single matrix equation Ax = b, and write out this matrix equation. 11. a. 2x₁ 9x1 X₁ + 5x₂ + 4x3 = b. 4x₁ 3x₂ + 5x3 = 7 x₂ + x3 = -1 0 - 3x3 + x₁ = 1 5X + X2 - 8x₁ = 3 2x1 - 5x₂ + 9x3 - X4 = 0 3x2 X3 + 7x₁ = 2
In each part of Exercises 11-12, find matrices A, x, and b that express the given linear system as a single matrix equation Ax = b, and write out this matrix equation. 11. a. 2x₁ 9x1 X₁ + 5x₂ + 4x3 = b. 4x₁ 3x₂ + 5x3 = 7 x₂ + x3 = -1 0 - 3x3 + x₁ = 1 5X + X2 - 8x₁ = 3 2x1 - 5x₂ + 9x3 - X4 = 0 3x2 X3 + 7x₁ = 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Systems and Matrix Equations
In each part of Exercises 11–12, find matrices \( \mathbf{A} \), \( \mathbf{x} \), and \( \mathbf{b} \) that express the given linear system as a single matrix equation \( \mathbf{Ax} = \mathbf{b} \), and write out this matrix equation.
#### Exercise 11
**a.**
Given the linear system:
\[
\begin{aligned}
2x_1 - 3x_2 + 5x_3 &= 7 \\
9x_1 - x_2 + x_3 &= -1 \\
x_1 + 5x_2 + 4x_3 &= 0
\end{aligned}
\]
Rewrite this system in matrix form \( \mathbf{Ax} = \mathbf{b} \) where:
\[
\mathbf{A} = \begin{pmatrix}
2 & -3 & 5 \\
9 & -1 & 1 \\
1 & 5 & 4 \\
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix}
7 \\
-1 \\
0 \\
\end{pmatrix}
\]
Thus, the matrix equation is:
\[
\begin{pmatrix}
2 & -3 & 5 \\
9 & -1 & 1 \\
1 & 5 & 4 \\
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{pmatrix} =
\begin{pmatrix}
7 \\
-1 \\
0 \\
\end{pmatrix}
\]
**b.**
Given the linear system:
\[
\begin{aligned}
4x_1 - 3x_3 + x_4 &= 1 \\
5x_1 + x_2 - 8x_4 &= 3 \\
2x_1 - 5x_2 + 9x_3 - x_4 &= 0 \\
3x_2 - x_3 + 7x_4 &= 2
\end{aligned}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2F90013816-db17-493f-b69d-cea261b32659%2F10fu0t_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Systems and Matrix Equations
In each part of Exercises 11–12, find matrices \( \mathbf{A} \), \( \mathbf{x} \), and \( \mathbf{b} \) that express the given linear system as a single matrix equation \( \mathbf{Ax} = \mathbf{b} \), and write out this matrix equation.
#### Exercise 11
**a.**
Given the linear system:
\[
\begin{aligned}
2x_1 - 3x_2 + 5x_3 &= 7 \\
9x_1 - x_2 + x_3 &= -1 \\
x_1 + 5x_2 + 4x_3 &= 0
\end{aligned}
\]
Rewrite this system in matrix form \( \mathbf{Ax} = \mathbf{b} \) where:
\[
\mathbf{A} = \begin{pmatrix}
2 & -3 & 5 \\
9 & -1 & 1 \\
1 & 5 & 4 \\
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix}
7 \\
-1 \\
0 \\
\end{pmatrix}
\]
Thus, the matrix equation is:
\[
\begin{pmatrix}
2 & -3 & 5 \\
9 & -1 & 1 \\
1 & 5 & 4 \\
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{pmatrix} =
\begin{pmatrix}
7 \\
-1 \\
0 \\
\end{pmatrix}
\]
**b.**
Given the linear system:
\[
\begin{aligned}
4x_1 - 3x_3 + x_4 &= 1 \\
5x_1 + x_2 - 8x_4 &= 3 \\
2x_1 - 5x_2 + 9x_3 - x_4 &= 0 \\
3x_2 - x_3 + 7x_4 &= 2
\end{aligned}
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