Question

If A, B are two n × n non-singular matrices, then
(a) AB is non-singular
(b) AB is singular
(c) ${\left(AB\right)}^{-1}{A}^{-1}{B}^{-1}$
(d) (AB)⁻¹ does not exist

Answer 1

(a) AB is non-singular

$$\begin{gathered} A\mathrm{and}B\mathrm{are}\mathrm{non}-\mathrm{singular}\mathrm{matrices}\mathrm{of}\mathrm{order}n\times n. \\ \therefore \left|\mathrm{A}\right|\ne 0\mathrm{and}\left|\mathrm{B}\right|\ne 0...\left(1\right) \\ A\mathit{}\mathrm{and}B\mathrm{are}\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{order},\mathrm{so}AB\mathrm{is}\mathrm{defined}\mathrm{and}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{order}. \\ \mathrm{Thus}, \\ \left|\mathrm{AB}\right|=\left|\mathrm{A}\right|\left|\mathrm{B}\right| \\ \Rightarrow \left|\mathrm{AB}\right|\ne 0\left[\mathrm{Using}\left(1\right)\right] \\ \mathrm{Thus},AB\mathrm{is}\mathrm{non}-\mathrm{singular}. \end{gathered}$$