The inverse of the matrix \(A = \begin{bmatrix}\frac{1}{2}&0&0\\0&\frac{1}{4}&0\\0&0&\frac{1}{8}\end{bmatrix}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}2&0&0\\0&4&0\\0&0&8\end{bmatrix}\)
The inverse of a diagonal matrix with non-zero diagonal entries is the diagonal matrix of the reciprocals of those entries:
\( \displaystyle A^{-1} = \begin{bmatrix}2&0&0\\0&4&0\\0&0&8\end{bmatrix} \)
Check: each diagonal entry of \(AA^{-1}\) is a product such as \(\frac{1}{2} \times 2 = 1\), and every other entry is 0, so \(AA^{-1} = I\).
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{64}\begin{bmatrix}2&0&0\\0&4&0\\0&0&8\end{bmatrix}\) — Incorrect. This is \(\frac{1}{64}A^{-1}\); its product with A is \(\frac{1}{64}I\), not I.
Option (2) \(\displaystyle\frac{1}{64}\begin{bmatrix}\frac{1}{2}&0&0\\0&\frac{1}{4}&0\\0&0&\frac{1}{8}\end{bmatrix}\) — Incorrect. This is \(\frac{1}{64}A\); its product with A has first entry \(\frac{1}{64} \times \frac{1}{4} = \frac{1}{256}\), not 1.
Option (4) \(\displaystyle\frac{1}{64}\begin{bmatrix}\frac{1}{32}&0&0\\0&\frac{1}{16}&0\\0&0&\frac{1}{8}\end{bmatrix}\) — Incorrect. Here \(|A| = \frac{1}{64}\) and \(\mathrm{adj}\,A\) is the matrix shown, so this option is \(|A|\,\mathrm{adj}\,A\). The formula is \(A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\): adj A must be divided by \(|A|\), which gives \(64\,\mathrm{adj}\,A\), the matrix of Option (3).