The inverse of the matrix \(\begin{bmatrix}4&0&0\\0&5&0\\0&0&6\end{bmatrix}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}1/4&0&0\\0&1/5&0\\0&0&1/6\end{bmatrix}\)
The inverse of a diagonal matrix with non-zero diagonal entries is the diagonal matrix of their reciprocals. Check by multiplication:
\( \displaystyle \begin{bmatrix}4&0&0\\0&5&0\\0&0&6\end{bmatrix}\begin{bmatrix}\frac{1}{4}&0&0\\0&\frac{1}{5}&0\\0&0&\frac{1}{6}\end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} = I \)
So the inverse is the matrix in Option (3).