If $ A = \left[\begin{array}{cc} 3 & 1 \\2 & 4 \end{array}\right] $, then the determinant of the adjoint of $ A^2 $ is:
100
To determine the determinant of the adjoint of \( A^2 \) for the matrix \( A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \), proceed as follows:
Step 1: Compute \( A^2 \)
The square of matrix \( A \) is defined as \( A^2 = A \times A \).
Computation:
\[ A^2 = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \times \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 3 + 1 \times 2 & 3 \times 1 + 1 \times 4 \\ 2 \times 3 + 4 \times 2 & 2 \times 1 + 4 \times 4 \end{bmatrix} = \begin{bmatrix} 11 & 7 \\ 14 & 18 \end{bmatrix} \]
Step 2: Determine the adjoint of \( A^2 \)
For a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), its adjoint is \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Given \( A^2 = \begin{bmatrix} 11 & 7 \\ 14 & 18 \end{bmatrix} \), the adjoint is:
\( \text{adj}(A^2) = \begin{bmatrix} 18 & -7 \\ -14 & 11 \end{bmatrix} \)
Step 3: Calculate the determinant of the adjoint
The determinant of a matrix \( \begin{bmatrix} e & f \\ g & h \end{bmatrix} \) is computed as \( eh - fg \).
For \( \text{adj}(A^2) = \begin{bmatrix} 18 & -7 \\ -14 & 11 \end{bmatrix} \), the calculation is:
\(\det(\text{adj}(A^2)) = 18 \times 11 - (-7) \times (-14) = 198 - 98 = 100\)