Question:
If
\[
A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}
\]
then \( A^{-1} \) is:
If
\[
A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}
\]
then \( A^{-1} \) is:
Show Hint
For a diagonal matrix \( A = {diag}(a_1, a_2, \dots, a_n) \), the inverse is \( A^{-1} = {diag}\left(\frac{1}{a_1}, \frac{1}{a_2}, \dots, \frac{1}{a_n}\right) \), provided \( a_i \neq 0 \).
Updated On: Jan 13, 2026
- \[ \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix} \]
- \[ 30 \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix} \]
- \[ \frac{1}{30} \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]
- \[ \frac{1}{30} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Show Solution
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The Correct Option is A
Solution and Explanation
The inverse of a diagonal matrix is obtained by taking the reciprocal of the diagonal elements. For
\[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}, \]the diagonal elements are 2, 3, and 5. Thus:
\[ A^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix}. \]This matches option (A).
Final Answer: \( \boxed{(A)} \)Was this answer helpful?
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