Exercise 1.2.6

If $B=I$, then the rank-one matrices $a_1{b}_1^{\text{∗}}=\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$, $a_2{b}_2^{\text{∗}}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill a_2\hfill & \hfill 0\hfill \end{array}\right]$and $a_3{b}_3^{\text{∗}}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill a_3\hfill \end{array}\right]$, they add up to $\mathit{AI}=A$.