Question

Let $A$ be a square matrix of order 3 such that $adj(adj(adj(A)))=\begin{bmatrix} 16 & 0 & 4 \\ 5 & 4 & 0 \\ 1 & 4 & 3 \end{bmatrix}$ and det$(A)$ is positive, then which of the following must be correct?

• A. $8 \cdot \text{trace}(A^{-1}) = 32$
• B. $\det(\text{adj } A) = 4$
• C. $8 \cdot \text{trace}(A^{-1}) = 23$
• D. $\det(\text{adj } A) = 2$

Answer 1

Answer: (B), (C)

Options 2 and 3 are correct.

Answer 2

Using the identities for adjugate, we show that $adj(adj(adj A)) = (\det A)^3A^{-1}$. Equating this with the given matrix, we obtain $\text{trace}(A^{-1}) = \frac{23}{(\det A)^3}$ and, via computing the determinant of the given matrix, deduce that $\det A = 2$. Hence, $\det(\text{adj } A) = 4$ and $8 \cdot \text{trace}(A^{-1}) = 23$.