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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics5 Marks
Question
If $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, verify that $A(\operatorname{adj} A)=(\operatorname{adj} A) A=|A|$.
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Answer

$\text { For } A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] $
$A _{11}=(-1)^{1+1}(4)=4$
$A_{12}=(-1)^{1+2}(3)=-3$
$A_{21}=(-1)^{2+1}(2)=-2$
$A_{22}=(-1)^{2+2}(1)=1$
$ \operatorname{adj} A=\left[\begin{array}{ll} A_{11} & A_{21} \\ A_{12} &  A_{22}\end{array}\right] \quad=\left[\begin{array}{cc} 4 & -2 \\ -3 & 1 \end{array}\right] $
$ \therefore \quad A (\operatorname{adj} A )=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\left[\begin{array}{cc} 4 & -2 \\ -3 & 1 \end{array}\right]=\left[\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right] $
$ (\operatorname{adj} A) \cdot A=\left[\begin{array}{cc} 4 & -2 \\ -3 & 1 \end{array}\right]\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $
$ =\left[\begin{array}{cc} 4-6 & 8-8 \\ -3+3 & -6+4 \end{array}\right] $
$ =\left[\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right] $ and $|A| 1=\left|\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right|\left[\begin{array}{ll} 1 & 0 \\ 0 & 1
\end{array}\right] $
$ =(-2)\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ =\left[\begin{array}{cc} -2 & 0 \\
0 & -2 \end{array}\right] $
From $(i), (ii)$ and $(iii)$ we get, $A (\operatorname{adj} A )=(\operatorname{adj} A ) A =| A | I$
$($Note that this equation is valid for every nonsingular square matrix $A)$

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