Find the matrix X satisfying the equation: 2 & 1 5 & 3 X 5 & 3 3 … - Vidyadip

Question

Find the matrix $X$ satisfying the equation:
$\begin{bmatrix}2 & 1 \\5 & 3 \end{bmatrix}\text{X}\begin{bmatrix}5 & 3 \\3 & 2 \end{bmatrix}=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}.$

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Answer

Let $\text{A}=\begin{bmatrix}2 & 1 \\5 & 3 \end{bmatrix}$
$\text{B}=\begin{bmatrix}5 & 3 \\3 & 2 \end{bmatrix}$
Then the given equation can be wirtten as
$A \times B = I$
$\Rightarrow X = A^{-1}B^{-1}$
Now $|A| = 6 - 5 = 1$
$|B| = 10 - 9 = 1$
$\text{A}^{-1}=\frac{\text{adj (A)}}{|\text{A}|}=\begin{bmatrix}3 & -1 \\-5 & 2 \end{bmatrix}$
$\text{B}^{-1}=\frac{\text{adj (B)}}{|\text{B}|}=\begin{bmatrix}2 & -3 \\-3 & 5 \end{bmatrix}$
$\therefore\ \text{X}=\begin{bmatrix} 3 & -1 \\-5 & 2 \end{bmatrix}\begin{bmatrix} 2 & -3 \\-3 & 2 \end{bmatrix}$
$=\begin{bmatrix} 9 & -14 \\-16 & 25 \end{bmatrix}$

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