Question

If $ A = \left[\begin{array}{cc} 3 & 1 \\2 & 4 \end{array}\right] $, then the determinant of the adjoint of $ A^2 $ is:

• A. 121
• B. 144
• C. 100
• D. 196

Hint:

Key Fact: For a 2x2 matrix, \( \det(\text{adj}((A)) = \det((A) \), and \( \det(A^2) = (\det((A))^2 \).

Answer 1

The Correct Option is C

To find the determinant of the adjoint of \( A^2 \) for the matrix \( A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \), follow these steps: 

Step 1: Calculate \( A^2 \)

The square of a matrix \( A \) is given by \( A^2 = A \times A \).

Calculate:

\[ A^2 = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \times \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 3 + 1 \times 2 & 3 \times 1 + 1 \times 4 \\ 2 \times 3 + 4 \times 2 & 2 \times 1 + 4 \times 4 \end{bmatrix} = \begin{bmatrix} 11 & 7 \\ 14 & 18 \end{bmatrix} \]

Step 2: Find the adjoint of \( A^2 \)

The adjoint of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).

For \( A^2 = \begin{bmatrix} 11 & 7 \\ 14 & 18 \end{bmatrix} \), 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 given by \( eh - fg \).

For \( \text{adj}(A^2) = \begin{bmatrix} 18 & -7 \\ -14 & 11 \end{bmatrix} \), calculate:

\(\det(\text{adj}(A^2)) = 18 \times 11 - (-7) \times (-14) = 198 - 98 = 100\)

Answer 2

To solve the problem, we need to find the determinant of the adjoint of $A^2$ given the matrix: 

$ A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} $

1. Find $\det(A)$:
$\det(A) = (3)(4) - (1)(2) = 12 - 2 = 10$

2. Use the property of determinants:
$\det(A^2) = (\det(A))^2 = 10^2 = 100$

3. Relation between determinant of adjoint and determinant of matrix:
For an $n \times n$ matrix $M$, $\det(\text{adj}(M)) = (\det(M))^{n-1}$

Here, $n = 2$, so
$\det(\text{adj}(A^2)) = (\det(A^2))^{2-1} = \det(A^2) = 100$

Final Answer:
The determinant of the adjoint of $A^2$ is $ {100} $.