Question
Find matrix \(A\) such that \[ \begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{bmatrix} A = \begin{bmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{bmatrix} \]
Solution
Step 1: Assume \(A\)
\[ A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \]Step 2: Multiply
\[ \begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} = \begin{bmatrix} 2a – d & 2b – e & 2c – f \\ a & b & c \\ -3a + 4d & -3b + 4e & -3c + 4f \end{bmatrix} \]Step 3: Compare
\[ \begin{bmatrix} 2a – d & 2b – e & 2c – f \\ a & b & c \\ -3a + 4d & -3b + 4e & -3c + 4f \end{bmatrix} = \begin{bmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{bmatrix} \]Step 4: Solve
From second row: \[ a=1,\quad b=-2,\quad c=-5 \] Substitute: \[ 2(1) – d = -1 \Rightarrow d=3 \] \[ 2(-2) – e = -8 \Rightarrow e=4 \] \[ 2(-5) – f = -10 \Rightarrow f=0 \] (Check third row → consistent)Final Answer
\[
A =
\begin{bmatrix}
1 & -2 & -5 \\
3 & 4 & 0
\end{bmatrix}
\]