Let \(A = \begin{bmatrix}2 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 4\end{bmatrix}\). Consider the following statements:
(A) \(|A| = -8\)
(B) \(A^{-1} = \begin{bmatrix}\frac{1}{2} & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & \frac{1}{4}\end{bmatrix}\)
(C) \(\mathrm{adj}\,A = \begin{bmatrix}-4 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & -2\end{bmatrix}\)
(D) \(|\mathrm{adj}\,A| = -64\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
The determinant of a diagonal matrix is the product of its diagonal elements:
\(|A| = 2 \times (-1) \times 4 = -8\)
All off-diagonal cofactors are 0. The diagonal cofactors are:
\(A_{11} = (-1)(4) = -4\)
\(A_{22} = (2)(4) = 8\)
\(A_{33} = (2)(-1) = -2\)
\(\displaystyle \mathrm{adj}\,A = \begin{bmatrix}-4 & 0 & 0\\ 0 & 8 & 0\\ 0 & 0 & -2\end{bmatrix}\)
\(\displaystyle A^{-1} = \frac{1}{-8}\,\mathrm{adj}\,A = \begin{bmatrix}\frac{1}{2} & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & \frac{1}{4}\end{bmatrix}\)
\(|\mathrm{adj}\,A| = |A|^{2} = 64\)
So \(|A| = -8\), adj A = diag(−4, 8, −2), \(A^{-1}\) = diag(1/2, −1, 1/4) and \(|\mathrm{adj}\,A| = 64\).
Checking the statements
(A) \(|A| = -8\) — Correct. \(2 \times (-1) \times 4 = -8\).
(B) \(A^{-1}\) = diag(1/2, −1, 1/4) — Correct. Each diagonal element is replaced by its reciprocal.
(C) adj A = diag(−4, 8, −2) — Correct. \(A_{11} = -4,\ A_{22} = 8,\ A_{33} = -2\).
(D) \(|\mathrm{adj}\,A| = -64\) — Incorrect. \((-4)(8)(-2) = 64 = |A|^{2}\), which is positive.