GATE 2015 EE – Question 13
If the sum of the diagonal elements of a $2 \times 2$ matrix is $-6$, then the maximum possible value of determinant of the matrix is ________.
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Correct answer: 9
Explanation
Write the matrix as $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ with $a + d = -6$. The determinant $ad - bc$ is largest when $bc = 0$ (taking a non-positive $bc$ at best), and $ad$ with $a + d = -6$ is largest at $a = d = -3$, giving $9$.