Question

If A is a singular matrix, then adj A is ______.

Options

• non-singular

• singular

• symmetric

• not defined

Answer 1

If A is a singular matrix, then adj A is singular.

\[\text{ A is singular, so } \left| A \right| = 0\].

By definition, we have

A \[adj \left( A \right) = O \]

\[ \Rightarrow \left| A adj \left( A \right) \right| = \left| O \right|\]

\[ \Rightarrow \left| A \right| \left| adj \left( A \right) \right| = 0 \]

\[ \Rightarrow \left| adj \left( A \right) \right| = 0 \]

\[\text{ Hence, }adj \left( A \right)\text{ is singular}\].