Create a system of linear equations A = such that the matrix A is a 2x2 matrix with a determinant of 11 and 7 = (² system of equations.
Create a system of linear equations A = such that the matrix A is a 2x2 matrix with a determinant of 11 and 7 = (² system of equations.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.2: Guassian Elimination And Matrix Methods
Problem 93E
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![**Problem Statement:**
Create a system of linear equations \(A\vec{x} = \vec{b}\) such that the matrix \(A\) is a 2x2 matrix with a determinant of 11 and \(\vec{b} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}\). Then, solve your system of equations.
**Step-by-Step Solution:**
1. **Define the Matrix \(A\):**
Let \(A\) be a 2x2 matrix:
\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
2. **Determinant Condition:**
The determinant of \(A\) is given by:
\[
\text{det}(A) = ad - bc = 11
\]
3. **Define the Vector \(\vec{b}\):**
Given:
\[
\vec{b} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}
\]
4. **System of Equations:**
We need to find \( \vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \) such that:
\[
A \vec{x} = \vec{b}
\]
This translates to:
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}
\]
Which gives us two linear equations:
\[
ax_1 + bx_2 = 22
\]
\[
cx_1 + dx_2 = 33
\]
5. **Solve the System:**
To solve for \(x_1\) and \(x_2\), we can use various methods such as substitution, elimination, or matrix inverses. Let's use the inverse method for simplicity:
\[
\vec{x} = A^{-1} \vec{b}
\]
6. **Find the Inverse of \(A\):**
Assuming \(A\) has an inverse, and knowing that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe362d5a9-bcf1-4bcc-9808-2899b06de487%2F63e5fc53-d980-4d2b-96ae-9113cbbd76cd%2Fi9bvrdo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Create a system of linear equations \(A\vec{x} = \vec{b}\) such that the matrix \(A\) is a 2x2 matrix with a determinant of 11 and \(\vec{b} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}\). Then, solve your system of equations.
**Step-by-Step Solution:**
1. **Define the Matrix \(A\):**
Let \(A\) be a 2x2 matrix:
\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
2. **Determinant Condition:**
The determinant of \(A\) is given by:
\[
\text{det}(A) = ad - bc = 11
\]
3. **Define the Vector \(\vec{b}\):**
Given:
\[
\vec{b} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}
\]
4. **System of Equations:**
We need to find \( \vec{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \) such that:
\[
A \vec{x} = \vec{b}
\]
This translates to:
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 22 \\ 33 \end{pmatrix}
\]
Which gives us two linear equations:
\[
ax_1 + bx_2 = 22
\]
\[
cx_1 + dx_2 = 33
\]
5. **Solve the System:**
To solve for \(x_1\) and \(x_2\), we can use various methods such as substitution, elimination, or matrix inverses. Let's use the inverse method for simplicity:
\[
\vec{x} = A^{-1} \vec{b}
\]
6. **Find the Inverse of \(A\):**
Assuming \(A\) has an inverse, and knowing that
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