Exercise 4.3.1

A matrix is invertible when it has no zero eigenvalues, so for $A\oplus B$ to be invertible, we want ${\lambda }_i+{\mu }_j\ne 0$ for every $i,j$. If (eigenvalue of $A$) = - (eigenvalue of $B$), then $A\oplus B$ is not invertible, its rank is less than $n^2$.

Example: $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$, and $B=\left[\begin{array}{cc}\hfill -4\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill -1\hfill \end{array}\right]$

Then we have $A\oplus B=A\otimes I_2+I_2\otimes B=\left[\begin{array}{cc}\hfill I_2\hfill & \hfill 2I_2\hfill \\ \hfill 3I_2\hfill & \hfill 4I_2\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill B\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill I_2+B\hfill & \hfill 2I_2\hfill \\ \hfill 3I_2\hfill & \hfill 4I_2+B\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill -3\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 3\hfill & \hfill 3\hfill & \hfill 3\hfill \end{array}\right]$

which is rank -3.