GATE DS&AI 2024 | Question: 25

Question

Consider the $3 \times 3$ matrix $\boldsymbol{M}=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6\end{array}\right]$.

The determinant of $\left(\boldsymbol{M}^{2}+12 \boldsymbol{M}\right)$ is $\_\_\_\_\_\_\_\_\_$.

Comments

Since row transformation $R_3 \leftarrow R_3-(R_1+R_2)$ makes $\det(M)=0$

Hence, $\det(M^2+12M)=\det(M(M+12I))=\det(M)\det(M+12I)=0$

 $\because \det(AB)=\det(A)\det(B)$ where A and B are square matrices of the same order.

Answer 1

determinant is 0