Question

If $A$ is a singular matrix, then $adj\left(A\right)$ is

• A. non-singular
• B. singular
• C. symmetric
• D. not defined

Answer 1

The correct option is B singular

Let A be of the order $n\times n$

Since A adjA$=\left|A\right|I$

Therefore $=|A.adjA|={\left|A\right|}^n$

$=\left|A\right|\left|adjA\right|={\left|A\right|}^n\Rightarrow \left|adjA\right|={\left|A\right|}^{n-1}$

Since A is singular then $\left|A\right|=0$

$\left|adjA\right|=0$ so adj A is also singular.