Chapter 3.2, Problem 42E

Let A = $\left[\begin{array}{cc}\text{1}& \text{0}\\ \text{0}& \text{1}\end{array}\right]$ and B = $\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right]$. Show that det(A + B) = det A + det B if and only if a + d = 0.