Question
Find \(A\) such that \[ [2\ \ 1\ \ 3] \begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} = A \]
Solution
Step 1: Single Matrix Equation
\[ [2\ \ 1\ \ 3] \begin{bmatrix} -1(1)+0+(-1)(-1) \\ -1(1)+0+0 \\ 0+0+1(-1) \end{bmatrix} = [2\ \ 1\ \ 3] \begin{bmatrix} 0 \\ -1 \\ -1 \end{bmatrix} \]Step 2: Multiply Row Matrix
\[ = 2\cdot0 + 1(-1) + 3(-1) \]Step 3: Simplify
\[ = -1 – 3 = -4 \]Final Answer
\[
A = -4
\]