Question

If \[ A = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}, \] then \( A^3 \) is:

• A. \(  \begin{bmatrix} 125 & 0 \\ 0 & 125 \end{bmatrix} \)
• B. \( \begin{bmatrix} 0 & 125 \\ 0 & 125 \end{bmatrix} \)
• C. \( \begin{bmatrix} 15 & 0 \\ 0 & 15 \end{bmatrix} \)
• D. \( \begin{bmatrix} 5^3 & 0 \\ 0 & 5^3 \end{bmatrix} \)

Hint:

For diagonal matrices, raising the matrix to a power involves raising each of the diagonal elements to that power. Off-diagonal elements remain zero.

Answer 1

The Correct Option is A

The matrix \( A \) is a diagonal matrix. To calculate \( A^3 \), we raise each diagonal element to the third power. Therefore, for \( A = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \), we have \( A^3 = \begin{bmatrix} 5^3 & 0 \\ 0 & 5^3 \end{bmatrix} = \begin{bmatrix} 125 & 0 \\ 0 & 125 \end{bmatrix} \). This result matches option (A).