Question:
If
\[
A = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix},
\]
then \( A^3 \) is:
If
\[
A = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix},
\]
then \( A^3 \) is:
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For diagonal matrices, raising the matrix to a power involves raising each of the diagonal elements to that power. Off-diagonal elements remain zero.
Updated On: Jan 13, 2026
\( \begin{bmatrix} 125 & 0 \\ 0 & 125 \end{bmatrix} \)
- \( \begin{bmatrix} 0 & 125 \\ 0 & 125 \end{bmatrix} \)
- \( \begin{bmatrix} 15 & 0 \\ 0 & 15 \end{bmatrix} \)
- \( \begin{bmatrix} 5^3 & 0 \\ 0 & 5^3 \end{bmatrix} \)
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The Correct Option is A
Solution and Explanation
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).
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