In section 4.2.
33) If $P_1$ projects onto $S$ and $P_2$ onto $T$, for $P_1 P_2=P_2 P_1$ to hold, the solution says that one of the subspaces has to contain the other. Wouldn't this also hold if they are e.g. orthogonal and $P_1 P_2=P_2P_1=0$ or share some non-zero intersection like to planes in $3d$ sharing a line in which the resulting vector after both projections is the same?
34) Proof: "If $A$ has $r$ independent columns and $B$ has $r$ independent rows, $AB$ is invertible". If $A$ is $m \times r$ and $B$ is $r \times n$, the resulting matrix is $m \times n$. Without further constrains this need not even be square, so why should $AB$ be invertible?