Question
Find matrix \(A\) such that \[ \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix} \]
Solution
Step 1: Understand Structure
Let \[ A = [a\ \ b\ \ c] \] (1 × 3 row matrix)Step 2: Multiply
\[ \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} [a\ \ b\ \ c] = \begin{bmatrix} 4a & 4b & 4c \\ a & b & c \\ 3a & 3b & 3c \end{bmatrix} \]Step 3: Compare
\[ \begin{bmatrix} 4a & 4b & 4c \\ a & b & c \\ 3a & 3b & 3c \end{bmatrix} = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix} \]Step 4: Solve
From second row: \[ a=-1,\quad b=2,\quad c=1 \] (Check: first row → \(4a=-4\), third row → \(3a=-3\), consistent)Final Answer
\[
A = [-1\ \ 2\ \ 1]
\]