5 [2 points] Consider the system of linear equations
\[
\begin{array}{r}
2 x_{2}+2 x_{3}+x_{4}=1 \\
x_{1}-x_{3}=1 \\
-x_{1}+x_{2}+x_{4}=0 \\
2 x_{1}+x_{3}+x_{4}=1 .
\end{array}
\]
Rewrite this system as a matrix equation \( A \mathbf{x}=\mathbf{b} \), and find the determinant of \( \mathrm{A} \).

Answer

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Answer

\( \det(A) = 6 \)

Steps

Step 1 ：\(A = \begin{pmatrix} 0 & 2 & 2 & 1 \\ 1 & 0 & -1 & 0 \\ -1 & 1 & 0 & 1 \\ 2 & 0 & 1 & 1 \end{pmatrix}, \mathbf{x} = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{pmatrix}, \mathbf{b} = \begin{pmatrix} 1 \\ 1 \\ 0 \\ 1 \end{pmatrix} \)

Step 2 ：\( \det(A) = \begin{vmatrix} 0 & 2 & 2 & 1 \\ 1 & 0 & -1 & 0 \\ -1 & 1 & 0 & 1 \\ 2 & 0 & 1 & 1 \end{vmatrix} \)

Step 3 ：\( \det(A) = 6 \)

5 [2 points] Consider the system of linear equations\[\begin{array}{r}2 x_{2}+2 x_{3}+x_{4}=1 \\x_{1}-x_{3}=1 \\-x_{1}+x_{2}+x_{4}=0 \\2 x_{1}+x_{3}+x_{4}=1 .\end{array}\]Rewrite this system as a matrix equation \( A \mathbf{x}=\mathbf{b} \), and find the determinant of \( \mathrm{A} \).

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5 [2 points] Consider the system of linear equations
\[
\begin{array}{r}
2 x_{2}+2 x_{3}+x_{4}=1 \\
x_{1}-x_{3}=1 \\
-x_{1}+x_{2}+x_{4}=0 \\
2 x_{1}+x_{3}+x_{4}=1 .
\end{array}
\]
Rewrite this system as a matrix equation \( A \mathbf{x}=\mathbf{b} \), and find the determinant of \( \mathrm{A} \).