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

We are given the matrix \( A \) which is a diagonal matrix. To calculate \( A^3 \), we can use the property that for diagonal matrices, the exponentiation of the matrix involves raising each diagonal element to the power individually. This means: \[ A^3 = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}^3. \] Now, for a diagonal matrix, we calculate the cube of each diagonal element: \[ A^3 = \begin{bmatrix} 5^3 & 0 \\ 0 & 5^3 \end{bmatrix} = \begin{bmatrix} 125 & 0 \\ 0 & 125 \end{bmatrix}. \] This shows that the matrix \( A^3 \) is equal to \( \begin{bmatrix} 125 & 0 \\ 0 & 125 \end{bmatrix} \), which is option (A).