Chapter 1, Problem 25SE

Partitioned matrices can be multiplied by the row-column rule just as if the matrix entries were numbers provided that the sizes of all matrices are such that the necessary operations can be performed. Thus, for example, if A is partitioned into a 2 × 2 matrix and B into a 2 × 1 matrix, then

$AB=\left[\begin{array}{l}{A}_{11}{A}_{12}\\ A_{21}{A}_{22}\end{array}\right]\left[\begin{array}{l}{B}_1\\ B_2\end{array}\right]=\left[\begin{array}{l}{A}_{11}{B}_1+A_{12}{B}_2\\ A_{21}{B}_1+A_{22}{B}_2\end{array}\right]$ (*)

provided that the sizes are such that AB, the two sums, and the four products are all defined.

Let A and B be the following partitioned matrices.

$B=\left[\begin{array}{l}30\\ 21\\ \underset{\_}{4-1}\\ 03\\ 25\end{array}\right]\text{}=\text{}\left[\begin{array}{l}{B}_1\\ B_2\end{array}\right]$

1. a. Confirm that the sizes of all matrices are such that the product AB can be obtained using Formula (∗).
2. b. Confirm that the result obtained using Formula (∗) agrees with that obtained using ordinary matrix multiplication.