Problem-9

Question

Is \(\text{det}(A + B) = \text{det}(A) + \text{det}(B)\)? If it is true, prove it. If not, give a counter example.

Hint

Think about \(I\) and \(-I\).

Solution

We have: \[ A = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix} \] The determinant of both matrices is \(1\). So \(\text{det}(A) + \text{det}(B) = 2\). However, \(A + B\) is the zero matrix, whose determinant is zero. It follows that \(\text{det}(A + B)\) need not necessarily be equal to \(\text{det}(A) + \text{det}(B)\). An even simpler example: \[ \text{det}(I + I) = \text{det}(2I) = 4 \] But \(2 \cdot \text{det}(I) = 2\).