Question

If A, B are two n × n non-singular matrices, then __________ .

Options

• AB is non-singular

• AB is singular

• \[\left( AB \right)^{- 1} A^{- 1} B^{- 1}\]

• (AB)⁻¹ does not exist

Answer 1

AB is non-singular

A and B are non - singular matrices of order n × n.
\[ \therefore \left| A \right| \neq 0\text{ and }\left| B \right| \neq 0 . . . \left( 1 \right)\]

A and B are of the same order, so AB is defined and is of the same order .

Thus, 
\[\left| AB \right| = \left| A \right|\left| B \right| \]
\[ \Rightarrow \left| AB \right| \neq 0 \left[\text{ Using }\left( 1 \right) \right] \]
Thus, AB is non - singular .