If \(x, y, z\) are non-zero numbers, then the inverse of matrix \(A = \begin{bmatrix}x&0&0\\0&y&0\\0&0&z\end{bmatrix}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}\displaystyle\frac{1}{x}&0&0\\0&\displaystyle\frac{1}{y}&0\\0&0&\displaystyle\frac{1}{z}\end{bmatrix}\)
Since \(A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\):
\(|A| = xyz \ne 0\)
All off-diagonal cofactors are 0, and the diagonal cofactors are \(yz\), \(xz\) and \(xy\):
\( \mathrm{adj}\,A = \begin{bmatrix}yz&0&0\\0&xz&0\\0&0&xy\end{bmatrix} \)
\( \displaystyle A^{-1} = \frac{1}{xyz}\begin{bmatrix}yz&0&0\\0&xz&0\\0&0&xy\end{bmatrix} = \begin{bmatrix}\frac{1}{x}&0&0\\0&\frac{1}{y}&0\\0&0&\frac{1}{z}\end{bmatrix} \)
Shortcut: The inverse of a diagonal matrix is the diagonal matrix of the reciprocals: multiplying A by Option (1) gives \(\mathrm{diag}\left(x \cdot \frac{1}{x}, y \cdot \frac{1}{y}, z \cdot \frac{1}{z}\right) = I\).