Question

If A is an invertible matrix of order 3, then which of the following is not true
(a) $\left|\mathrm{adj}A\right|={\left|A\right|}^2$
(b) ${\left(A^{-1}\right)}^{-1}=A$
(c) If $BA=CA,\mathrm{than}B\ne C$, where B and C are square matrices of order 3
(d) ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1},\mathrm{where}B\ne {\left[b_{ij}\right]}_{3\times 3}\mathrm{and}\left|B\right|\ne 0$

Answer 1

(c) If $BA=CA$, then $B\ne C$ where B and C are square matrices of order 3.

If A is an invertible matrix, then $A^{-1}$ exists.

Now,
$BA=CA$
On multiplying both sides by $A^{-1}$, we get
$BA{A}^{-1}=CA{A}^{-1}$

$$\begin{gathered} \Rightarrow BI=CI\left[\because A{A}^{-1}=I\mathrm{where}I\mathrm{is}\mathrm{the}\mathrm{identity}\mathrm{matrix}\right] \\ \Rightarrow B=C \end{gathered}$$

Therefore, the statement ​given in (c) is not true.