• # question_answer 72) Which of the following is correct? A)  If $A$ and $B$ are square matrices of order $3$ such that $|A|\,=-1,\,\,|B|\,=3$, then the determinant of $3\,\,AB$ is equal to$27$.B)  If $A$ is an invertible matrix, then $\det ({{A}^{-1}})$ is equal to$\det (A)$C)  If $A$ and $B$ are matrices of the same order, then${{(A+B)}^{2}}={{A}^{2}}+2AB+{{B}^{2}}$is possible if$AB=I$D)  None of these

[a] We have$|AB|\,\,=\,\,|A||B|$ Also for a square matrix of order$3,\,\,|kA|={{k}^{3}}|A|$ because each element of the matrix A is multiplied by $k$ and hence in this case we will have k3 common $\therefore$$|3AB|\,\,={{3}^{3}}|A||B|\,\,=27(-1)(3)=-81$ [b] Since $A$ is invertible, therefore ${{A}^{-1}}$ exists and             $A{{A}^{-1}}=I=\det (A{{A}^{-1}})=\det (I)$ $\Rightarrow$            $\det (A)\det ({{A}^{-1}})=1$ $\Rightarrow$            $\det ({{A}^{-1}})=\frac{1}{\det (A)}$ [c]${{(A+B)}^{2}}=(A+B)(A+B)$ $={{A}^{2}}+AB+BA+{{B}^{2}}$ $={{A}^{2}}+2AB+{{B}^{2}}$if$AB=BA$.