If `A=|{:(,2,-3),(,-4,1):}|` then adj `(3A^(2)+12A)` is equal to

Options

A. `[{:(,72,-84),(,-63,51):}]`
B. `[{:(,51,63),(,84,72):}]`
C. `[{:(,51,84),(,63,72):}]`
D. `[{:(,72,-63),(,-84,51):}]`

Answer

To solve the problem, we need to find the adjoint of the matrix expression \(3A^2 + 12A\) where \(A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}\).

### Step 1: Calculate \(A^2\)

First, we need to multiply matrix \(A\) by itself:

\[ A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix} \]

Calculating \(A^2 = A \cdot A\):

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

Using matrix multiplication:

- First element: \(2 \cdot 2 + (-3) \cdot (-4) = 4 + 12 = 16\)
- Second element: \(2 \cdot (-3) + (-3) \cdot 1 = -6 - 3 = -9\)
- Third element: \((-4) \cdot 2 + 1 \cdot (-4) = -8 - 4 = -12\)
- Fourth element: \((-4) \cdot (-3) + 1 \cdot 1 = 12 + 1 = 13\)

Thus,

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

### Step 2: Calculate \(3A^2\)

Now, we multiply \(A^2\) by 3:

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

### Step 3: Calculate \(12A\)

Next, we calculate \(12A\):

\[ 12A = 12 \cdot \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix} = \begin{pmatrix} 24 & -36 \\ -48 & 12 \end{pmatrix} \]

### Step 4: Calculate \(3A^2 + 12A\)

Now, we add \(3A^2\) and \(12A\):

\[ 3A^2 + 12A = \begin{pmatrix} 48 & -27 \\ -36 & 39 \end{pmatrix} + \begin{pmatrix} 24 & -36 \\ -48 & 12 \end{pmatrix} \]

Calculating the sum:

- First element: \(48 + 24 = 72\)
- Second element: \(-27 + (-36) = -63\)
- Third element: \(-36 + (-48) = -84\)
- Fourth element: \(39 + 12 = 51\)

Thus,

\[ 3A^2 + 12A = \begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix} \]

### Step 5: Calculate the adjoint of \(3A^2 + 12A\)

To find the adjoint of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), we use the formula:

\[ \text{adj} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Applying this to our matrix:

\[ \text{adj}(3A^2 + 12A) = \begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix} \]

### Final Result

Thus, the adjoint of \(3A^2 + 12A\) is:

\[ \text{adj}(3A^2 + 12A) = \begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix} \]