Question

If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), then the determinant of \( A^2 \) is:

• A. -4
• B. 4
• C. 25
• D. 49

Hint:

Determinant of power equals power of determinant.

Answer 1

The Correct Option is B

To solve this problem, we need to determine the determinant of the square of matrix \( A \), denoted as \( A^2 \). Given that:

\( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)

The first step is to find the square of matrix \( A \), which is \( A^2 = A \times A \).

\[ \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} (1 \times 1 + 2 \times 3) & (1 \times 2 + 2 \times 4) \\ (3 \times 1 + 4 \times 3) & (3 \times 2 + 4 \times 4) \end{pmatrix} \]

This simplifies to:

\[ = \begin{pmatrix} 1 + 6 & 2 + 8 \\ 3 + 12 & 6 + 16 \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 15 & 22 \end{pmatrix} \]

Next, we find the determinant of this resultant matrix \( A^2 \):

\[ \text{det}(A^2) = (7 \times 22) - (10 \times 15) \]

Calculating these products, we have:

\[ = 154 - 150 = 4 \]

Thus, the determinant of \( A^2 \) is \( 4 \), which is the correct answer.