## Linear Algebra and Its Applications, Exercise 2.5.14

Exercise 2.5.14. Suppose we have the block matrices $\begin{bmatrix} A&B \\ C&D \end{bmatrix} \begin{bmatrix} E&F \\ G&H \end{bmatrix}$

with the following properties:

• $A$ and $B$ have $m_1$ rows
• $C$ and $D$ have $m_2$ rows
• $E$ and $F$ have $n_1$ rows
• $G$ and $H$ have $n_2$ rows
• $E$ and $H$ are square matrices

What is the product of the two matrices, and what are the sizes of each of the product’s component submatrices?

Answer: Since $E$ is a square matrix with $n_1$ rows it must have $n_1$ columns. If $E$ has $n_1$ columns then $G$ must have $n_1$ columns also. Similarly, since $H$ is a square matrix with $n_2$ rows it must have $n_2$ columns, and $F$ must have $n_2$ columns as well.

In computing the (1, 1) element of  the product matrix we multiply $A$ by $E$ and $B$ by $G$. (See the post on multiplying block matrices.) This means that the number of columns of $A$ must equal the number of rows of $E$ or $n_1$, and the number of columns of $B$ must equal the number of rows of $G$ or $n_2$. The number of columns of $C$ and $D$ must then be $n_1$ and $n_2$ respectively.

The sizes of each of the elements $A$ through $H$ is thus as follows:

• $A$ is $m_1$ by $n_1$
• $B$ is $m_1$ by $n_2$
• $C$ is $m_2$ by $n_1$
• $D$ is $m_2$ by $n_2$
• $E$ is $n_1$ by $n_1$
• $F$ is $n_1$ by $n_2$
• $G$ is $n_2$ by $n_1$
• $H$ is $n_2$ by $n_2$

The product matrix is thus $\begin{bmatrix} A&B \\ C&D \end{bmatrix} \begin{bmatrix} E&F \\ G&H \end{bmatrix} = \begin{bmatrix} AE+BG&AF+BH \\ CE+DG&CF+DH \end{bmatrix}$

with the size of each element as follows:

• $AE+BG$ is $m_1$ by $n_1$
• $AF+BH$ is $m_1$ by $n_2$
• $CE+DG$ is $m_2$ by $n_1$
• $CF+DH$ is $m_2$ by $n_2$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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