If A = [ 2 − 4 − 3 1 ] , then adj ( 3 A 2 + 12 A ) is equal to, [ 72 − 84 − 63 51 ], [ 51 84 63 72 ], [ 72 − 63 − 84 51 ], [ 51 63 84 72 ]

adj ( 3 A 2 + 12 A ) = adj { 3 A ( A + 4 I )} = adj ( A + 4 I ) ⋅ adj ( 3 A ) = adj ( [ 6 − 4 − 3 5 ] ) ⋅ adj ( [ 6 − 12 − 9 3 ] ) = [ 5 4 3 6 ] [ 3 12 9 6 ] = [ 51 84 63 72 ]

If A = [ 2 − 4 − 3 1 ] , then adj ( 3 A 2 + 12 A ) is equal to

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If $A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$, then adj $\left(3 A^2+12 A\right)$ is equal to

MathematicsMatricesMHT CETMHT CET 2022 (06 Aug Shift 1)

Options:

A $\left[\begin{array}{cc}72 & -63 \\ -84 & 51\end{array}\right]$
B $\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]$
C $\left[\begin{array}{cc}72 & -84 \\ -63 & 51\end{array}\right]$
D $\left[\begin{array}{ll}51 & 84 \\ 63 & 72\end{array}\right]$

Solution:

2531 Upvotes Verified Answer

The correct answer is: $\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]$

$\begin{aligned} & \operatorname{adj}\left(3 \mathrm{~A}^2+12 \mathrm{~A}\right)=\operatorname{adj}\{3 \mathrm{~A}(\mathrm{~A}+4 \mathrm{I})\} \\ & =\operatorname{adj}(\mathrm{A}+4 \mathrm{I}) \cdot \operatorname{adj}(3 \mathrm{~A}) \\ & =\operatorname{adj}\left(\left[\begin{array}{cc}6 & -3 \\ -4 & 5\end{array}\right]\right) \cdot \operatorname{adj}\left(\left[\begin{array}{cc}6 & -9 \\ -12 & 3\end{array}\right]\right) \\ & =\left[\begin{array}{ll}5 & 3 \\ 4 & 6\end{array}\right]\left[\begin{array}{cc}3 & 9 \\ 12 & 6\end{array}\right] \\ & =\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]\end{aligned}$

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