Question:
If A is a singular matrix, then adj A is
(a) non-singular
(b) singular
(c) symmetric
(d) not defined
Solution:
(b) singular
$A$ is singular, so $|A|=0$.
By definition, we have
$A \operatorname{adj}(A)=O$
$\Rightarrow \mid A$ adj $(A)|=| O \mid$
$\Rightarrow|A||\operatorname{adj}(A)|=0$
$\Rightarrow|\operatorname{adj}(A)|=0$
Hence, adj $(A)$ is singular.
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