Question

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

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

Answer 1

The correct option is A

$\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$

Explanation for the correct option:

Calculating the adjoint for given expression:

The given matrix,

$A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$

Therefore,

$$\begin{gathered} 3A^2+12A=3\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]\times \left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]+12\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right] \\ =3\left[\begin{array}{cc}2\left(2\right)+(-3)(-4)& 2(-3)+(-3)\left(1\right)\\ (-4)\left(2\right)+1(-4)& -4(-3)+1\left(1\right)\end{array}\right]+12\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right] \\ =3\left[\begin{array}{cc}16& -9\\ -12& 13\end{array}\right]+12\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right] \\ =\left[\begin{array}{cc}3\times 16+12\times 2& 3(-9)+12(-3)\\ 3(-12)+12(-4)& 3\left(13\right)+12\left(1\right)\end{array}\right] \\ =\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right] \end{gathered}$$

Now, considering

$B=\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$

Therefore,

$Adj(3A^2+12A)=AdjB=Adj\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$

Since we know that

$Adj\left(B\right)=$Transpose of cofactor matrix of $B$

$$\begin{gathered} \Rightarrow Adj\left(B\right)={\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]}^T[\text{cofactor matrix of}B] \\ \Rightarrow Adj\left(B\right)=\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right] \end{gathered}$$

Hence, option (A) is the correct answer.