Consider the system of linear equations $$ \begin{array}{l}x+y-z=2 \ 3 x+2 y-z=3 \ -x-y+2 z=-1\end{array} $$ Write down the system in matrix form, $$ A X=b $$ where $$ X=\left[\begin{array}{l}x \ y \ z\end{array} ight] $$ Write down the coefficient matrix $$ A $$ of the above system. Determine the inverse $$ A^{-1} $$ of $$ A $$ by applying the matrix inverse algorithm, that is by applying the sequence of elementary row operations in the sense $$ \left[A \mid I_{3} ight] $$ where. $$ I_{3} $$ is the 3 by 3 identity matrix.

Question

Consider the system of linear equations $$ \begin{array}{l}x+y-z=2 \ 3 x+2 y-z=3 \ -x-y+2 z=-1\end{array} $$ Write down the system in matrix form, $$ A X=b $$ where $$ X=\left[\begin{array}{l}x \ y \ z\end{array}ight] $$ Write down the coefficient matrix $$ A $$ of the above system. Determine the inverse $$ A^{-1} $$ of $$ A $$ by applying the matrix inverse algorithm, that is by applying the sequence of elementary row operations in the sense $$ \left[A \mid I_{3}ight] $$ where. $$ I_{3} $$ is the 3 by 3 identity matrix.

Wells Tyler · Accepted Answer

The coefficient matrix $$A$$ for the system is: $$ A = \begin{bmatrix} 1 & 1 & -1 \ 3 & 2 & -1 \ -1 & -1 & 2 \end{bmatrix} $$ The inverse of $$A$$ is: $$ A^{-1} = \begin{bmatrix} -3 & 1 & -1 \ 5 & -1 & 2 \ 1 & 0 & 1 \end{bmatrix} $$

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