If $$A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$$, then Adj$$(3A^{2} + 12A)$$ is equal to:

We have the matrix

$$A=\begin{pmatrix}2 & -3\\-4 & 1\end{pmatrix}.$$

Our goal is to find $$\text{Adj}\,(3A^{2}+12A).$$ We shall proceed step by step, carrying out every algebraic operation in detail.

First, we need the square of $$A$$. For any two-by-two matrices, ordinary matrix multiplication rules apply:

$$A^{2}=A\cdot A=\begin{pmatrix}2 & -3\\-4 & 1\end{pmatrix} \begin{pmatrix}2 & -3\\-4 & 1\end{pmatrix}.$$

Multiplying the first row of the first matrix with the first column of the second gives

$$2\cdot2+(-3)\cdot(-4)=4+12=16.$$

Multiplying the first row with the second column gives

$$2\cdot(-3)+(-3)\cdot1=-6-3=-9.$$

Multiplying the second row with the first column gives

$$(-4)\cdot2+1\cdot(-4)=-8-4=-12.$$

Multiplying the second row with the second column gives

$$(-4)\cdot(-3)+1\cdot1=12+1=13.$$

So

$$A^{2}=\begin{pmatrix}16 & -9\\-12 & 13\end{pmatrix}.$$

Next, we need $$3A^{2}$$. Multiplying every entry of $$A^{2}$$ by $$3$$ gives

$$3A^{2}=3\begin{pmatrix}16 & -9\\-12 & 13\end{pmatrix} =\begin{pmatrix}48 & -27\\-36 & 39\end{pmatrix}.$$

Similarly, $$12A$$ is obtained by multiplying each entry of $$A$$ by $$12$$:

$$12A=12\begin{pmatrix}2 & -3\\-4 & 1\end{pmatrix} =\begin{pmatrix}24 & -36\\-48 & 12\end{pmatrix}.$$

Now we add these two matrices to obtain $$3A^{2}+12A$$:

$$3A^{2}+12A=\begin{pmatrix}48 & -27\\-36 & 39\end{pmatrix} +\begin{pmatrix}24 & -36\\-48 & 12\end{pmatrix} =\begin{pmatrix}48+24 & -27-36\\-36-48 & 39+12\end{pmatrix} =\begin{pmatrix}72 & -63\\-84 & 51\end{pmatrix}.$$

Let us denote this resulting matrix by $$B$$:

$$B=\begin{pmatrix}72 & -63\\-84 & 51\end{pmatrix}.$$

The next task is to find the adjugate (also called the adjoint) of a 2 × 2 matrix. For any matrix

$$\begin{pmatrix}a & b\\c & d\end{pmatrix},$$

the formula for the adjugate is

$$\text{Adj}\,\begin{pmatrix}a & b\\c & d\end{pmatrix} =\begin{pmatrix}d & -b\\-c & a\end{pmatrix}.$$

Applying this formula to $$B$$, we identify

$$a=72,\quad b=-63,\quad c=-84,\quad d=51.$$

Hence

$$\text{Adj}\,B =\begin{pmatrix}d & -b\\-c & a\end{pmatrix} =\begin{pmatrix}51 & -(-63)\\-(-84) & 72\end{pmatrix} =\begin{pmatrix}51 & 63\\84 & 72\end{pmatrix}.$$

This is exactly option B from the list provided.

Hence, the correct answer is Option 2.