Question
Find matrix \(A\) such that \[ A \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{bmatrix} \]
Solution
Step 1: Assume \(A\)
\[ A = \begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix} \]Step 2: Multiply
\[ A \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} a+4b & 2a+5b & 3a+6b \\ c+4d & 2c+5d & 3c+6d \\ e+4f & 2e+5f & 3e+6f \end{bmatrix} \]Step 3: Compare
\[ = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{bmatrix} \]Step 4: Solve Row-wise
First row: \[ a+4b=-7,\quad 2a+5b=-8 \Rightarrow b=-2,\ a=1 \] Second row: \[ c+4d=2,\quad 2c+5d=4 \Rightarrow d=0,\ c=2 \] Third row: \[ e+4f=11,\quad 2e+5f=10 \Rightarrow f=4,\ e=-5 \]Final Answer
\[
A =
\begin{bmatrix}
1 & -2 \\
2 & 0 \\
-5 & 4
\end{bmatrix}
\]