Nrite the given system of equations as a matrix equation and solve by using inverses. = k, 2x, + X2 - 4x, - X2 + X3 = k2 -X1 + X3 = k3
Nrite the given system of equations as a matrix equation and solve by using inverses. = k, 2x, + X2 - 4x, - X2 + X3 = k2 -X1 + X3 = k3
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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![### Solving Systems of Linear Equations Using Matrix Inverses
To solve the given system of linear equations using matrix inverses, we first express the system in matrix form.
#### System of Equations
The given system of equations is:
\[ \begin{align}
2x_1 + x_2 & = k_1 \\
-4x_1 - x_2 + x_3 & = k_2 \\
-x_1 + x_3 & = k_3 \\
\end{align} \]
#### Matrix Form
We can represent this system in the form \( AX = B \), where:
\[ A = \begin{pmatrix}
2 & 1 & 0 \\
-4 & -1 & 1 \\
-1 & 0 & 1
\end{pmatrix}, \quad
X = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}, \quad
B = \begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix} \]
Therefore, we can rewrite the system as:
\[ \begin{pmatrix}
2 & 1 & 0 \\
-4 & -1 & 1 \\
-1 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} =
\begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix} \]
#### Solving Using Inverses
To find \( X \), we need to multiply both sides of the equation by the inverse of matrix \( A \):
\[ X = A^{-1}B \]
First, we need to compute the inverse of matrix \( A \), denoted as \( A^{-1} \). Once we have \( A^{-1} \), we can multiply it with \( B \) to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).
#### Finding \( A^{-1} \)
Let us compute the inverse of \( A \):
\[ A^{-1} \begin{pmatrix}
a & b & c \\
d &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff60bf7ae-3e0a-47f1-9493-b686204b7388%2F53e31122-e12b-468a-8e89-302dbe2fbb38%2Fb3iakf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Linear Equations Using Matrix Inverses
To solve the given system of linear equations using matrix inverses, we first express the system in matrix form.
#### System of Equations
The given system of equations is:
\[ \begin{align}
2x_1 + x_2 & = k_1 \\
-4x_1 - x_2 + x_3 & = k_2 \\
-x_1 + x_3 & = k_3 \\
\end{align} \]
#### Matrix Form
We can represent this system in the form \( AX = B \), where:
\[ A = \begin{pmatrix}
2 & 1 & 0 \\
-4 & -1 & 1 \\
-1 & 0 & 1
\end{pmatrix}, \quad
X = \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}, \quad
B = \begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix} \]
Therefore, we can rewrite the system as:
\[ \begin{pmatrix}
2 & 1 & 0 \\
-4 & -1 & 1 \\
-1 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} =
\begin{pmatrix}
k_1 \\
k_2 \\
k_3
\end{pmatrix} \]
#### Solving Using Inverses
To find \( X \), we need to multiply both sides of the equation by the inverse of matrix \( A \):
\[ X = A^{-1}B \]
First, we need to compute the inverse of matrix \( A \), denoted as \( A^{-1} \). Once we have \( A^{-1} \), we can multiply it with \( B \) to find the values of \( x_1 \), \( x_2 \), and \( x_3 \).
#### Finding \( A^{-1} \)
Let us compute the inverse of \( A \):
\[ A^{-1} \begin{pmatrix}
a & b & c \\
d &
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