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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices3 Marks
Question
Find matrix $X$ such that $AX = B,$ where $A=\left[\begin{array}{cc}1 & -2 \\ -2 & 1\end{array}\right]$ and $B=\left[\begin{array}{c}-3 \\ -1\end{array}\right]$
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Answer

Let $\mathrm{X}=\left[\begin{array}{l}\boldsymbol{a} \\ b\end{array}\right]$ But $AX = B$
$\begin{aligned} & \therefore\left[\begin{array}{cc}1 & -2 \\ -2 & 1\end{array}\right]\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}-3 \\ -1\end{array}\right] \end{aligned} $
$ \therefore\left[\begin{array}{c}a-2 b \\ -2 a+b\end{array}\right]=\left[\begin{array}{l}-3 \\ -1\end{array}\right]$
By equality of matrices, we get $a – 2b = -3 …(i) -2a + b = -l …(ii)$
By $(i) x ^2 + (ii),$ we get $-3b =-7$
$\therefore b=\frac{7}{3}$
Substituting $b=\frac{7}{3}$ in $(i),$ we get
$a-2\left(\frac{7}{3}\right)=-3$
$\therefore a=-3+\frac{14}{3}=\frac{5}{3}$
$\therefore X=\left[\begin{array}{l}\frac{5}{3} \\ \frac{7}{3}\end{array}\right]$

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