Answer
True,false
Work Step by Step
(a) True. This is named in the section Addition and Scalar Multiplication. I say you try it for small matrices $A_{1}, A_{2}, B_{1},$ and $B_{2}$ if it doesn't seem obvious to you.
(b) False.we know that $A$ and $B$ can be multiplied iff the number of columns of $A$ is equal to the number of rows of $B .$ But the number of columns of $A$ depends on the number of columns of both $A_{11}$ and $A_{12}$ and the number of rows of $B$ depends on the number of rows of $B_{1}$ and $B_{2} .$ These don't have to be the same just since there are the same number of column partitions of $A$ as row partitions of $B$.
As a concrete counterexample, let
\[
A=\left[\begin{array}{cc|c}
1 & 2 & -4 \\
\hline 3 & 6 & -7 \\
4 & -5 & 1 / 2
\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\hline 2 & 2 & 2 & 2 \\
-2 & -2 & -2 & -2
\end{array}\right]
\]
We can explicitly see that though $A$ has exactly the same number of column partitions as $B$ has row partitions $(2), A$ DOESN'T have the same number of columns as $B$ has rows. So $A$ and $B$ cannot be multiplied as $A B$ (they also cannot be multiplied as $B A$ in this case, as you can easily $y$ confirm
