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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES2 Marks
Question
Given $\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix},$ compute $A^{-1}$ and show that $2A^{-1} = 9I – A$.
✓

Answer

$\text{A}=\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix}$
$\text{A}^{-1}=\frac{1}{|\text{A}|}\ \text{adj A}$
$=\frac{1}{14-12}\begin{bmatrix}7 & 3 \\4 & 2 \end{bmatrix}$
$=\frac{1}{2}\begin{bmatrix}7 & 3 \\4 & 2 \end{bmatrix}$
To Prove $2A^{-1} = 9I – A$
$LHS = 2A^{-1}$
$=2\times\frac{1}{2}\begin{bmatrix}7 & 3 \\4 & 2 \end{bmatrix}=\begin{bmatrix}7 & 3 \\4 & 2 \end{bmatrix}$
RHS = 9I – A
$=\begin{bmatrix}9 & 0 \\0 & 9 \end{bmatrix}-\begin{bmatrix}2 & -3 \\-4 & 7 \end{bmatrix}$
$=\begin{bmatrix}7 & 3 \\4 & 2 \end{bmatrix}$
LHS = RHS.

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