Question
Without using inverse, find \[ \begin{bmatrix} x & y \\ z & u \end{bmatrix} \] such that \[ \begin{bmatrix} 5 & -7 \\ -2 & 3 \end{bmatrix} \begin{bmatrix} x & y \\ z & u \end{bmatrix} = \begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix} \]
Solution
Step 1: Multiply Matrices
\[ = \begin{bmatrix} 5x – 7z & 5y – 7u \\ -2x + 3z & -2y + 3u \end{bmatrix} = \begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix} \]Step 2: Compare Elements
\[ 5x – 7z = -16 \] \[ -2x + 3z = 7 \] \[ 5y – 7u = -6 \] \[ -2y + 3u = 2 \]Step 3: Solve for \(x, z\)
\[ 5x – 7z = -16 \] \[ -2x + 3z = 7 \] Multiply second by 5 and first by 2: \[ -10x + 15z = 35 \] \[ 10x – 14z = -32 \] Add: \[ z = 3 \] \[ x = 1 \]Step 4: Solve for \(y, u\)
\[ 5y – 7u = -6 \] \[ -2y + 3u = 2 \] Multiply: \[ -10y + 15u = 10 \] \[ 10y – 14u = -12 \] Add: \[ u = -2 \] \[ y = -4 \]Final Answer
\[
\begin{bmatrix}
x & y \\
z & u
\end{bmatrix}
=
\begin{bmatrix}
1 & -4 \\
3 & -2
\end{bmatrix}
\]