Question

If $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then the value of |adj A| is

(a) a²⁷
(b) a⁹
(c) a⁶
(d) a²

Answer 1

(c) a⁶

$$\begin{gathered} A=\left[a00 \\ 0a0 \\ 00a\right] \\ \therefore \left|A\right|=\left|a00 \\ 0a0 \\ 00a\right|=a^3\ne 0 \\ \mathrm{and} \\ n=3 \\ \mathrm{Thus},\mathrm{we}\mathrm{have} \\ \left|adjA\right|={\left|A\right|}^{n-1}={\left(a^3\right)}^2=a^6 \end{gathered}$$