# Describe how to multiply matrices.

Question
Matrices
Describe how to multiply matrices.

2021-02-14
Step 1
Let's take an example to understand clearly
Let $$A=\begin{pmatrix}1 & 2\\3 & 4 \end{pmatrix} , B=\begin{pmatrix}4 & 6\\7 & 8 \end{pmatrix}$$
Step 2
$$A=\begin{pmatrix}1 & 2\\3 & 4 \end{pmatrix} , B=\begin{pmatrix}4 & 6\\7 & 8 \end{pmatrix}$$
$$AB=\begin{pmatrix}1 & 2\\3 & 4 \end{pmatrix}\begin{pmatrix}4 & 6\\7 & 8 \end{pmatrix}$$
Multiply the row of the first matrix by the columns of the second matrix
$$= \begin{pmatrix}1 \cdot 4 + 2 \cdot 7 &1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 4 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix}$$
$$=\begin{pmatrix}18 & 22\\40 & 50 \end{pmatrix}$$

### Relevant Questions

Describe how to subtract matrices.
Multiply the given matrix. After performing the multiplication, describe what happens to the elements in the first matrix. $$\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication.
$$\begin{bmatrix}1 & 4&3 \\0 & 1&4\\0&0&2 \end{bmatrix}\begin{bmatrix}3 & 2 \\1 & 1\\4&5 \end{bmatrix}$$
Multiply the following matrices:
$$A=\begin{bmatrix} -2 & 1 &5\\1 & 4&-5 \end{bmatrix} B=\begin{bmatrix}-1 & 7 \\2 & -2\\3&4 \end{bmatrix}$$ AB=?
Multiply the matrices:
$$\begin{bmatrix}1 & -1 &0\\2 & 1&3 \end{bmatrix} \begin{bmatrix}4 & -1 \\2 & 0\\1&1 \end{bmatrix}$$
Multiply the Following matrices:
$$\begin{bmatrix}2 & 3 \\1 & 0 \end{bmatrix} \times \begin{bmatrix}4 & 1 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}1 & 3 \\4 & 5\\1&2 \end{bmatrix} \times \begin{bmatrix}2 & -1&4 \\3 & 1&0 \end{bmatrix}$$
$$\begin{bmatrix}1 & 2&0 \\0 & 0&1\\0&0&0 \end{bmatrix}$$
(square roots of the identity matrix) For how many 2x2 matrices A is it true that $$A^2=I$$ ? Now answer the same question for n x n matrices where n>2