Question-35

Consider the \(3 \times 3\) matrix \(\mathbf{M} = \begin{bmatrix}1 & 2 & 3\\3 & 1 & 3\\4 & 3 & 6\end{bmatrix}\). The determinant of \(\mathbf{M}^{2} + 12 \mathbf{M}\) is ________ .

Hint

Add the first two rows.

Answer

\(0\)

Solution

Note the following: \[ \begin{aligned} \mathbf{M}^{2} + 12\mathbf{M} &= \mathbf{M}(\mathbf{M} + 12 \mathbf{I})\\\\ \implies \left|\mathbf{M}^2 + 12 \mathbf{M} \right| &= \left| \mathbf{M} \right| \cdot \left| \mathbf{M} + 12 \mathbf{I} \right|\\\\ \end{aligned} \] Let us first look at the determinant of \(\mathbf{M}\). Note that the third row of \(\mathbf{M}\) is equal to the sum of the first two rows. Hence, we see that \(\left| \mathbf{M} \right| = 0\). It follows that the answer is \(\boxed{0}\).