05. Matrix multiplication, 2

Examples

We're going to do some examples of matrix multiplication.

Example:

0.00 Consider the 90 degree rotation matrix $M=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ . We have

$M^{2}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$

$=\begin{pmatrix}-1&0\\ 0&-1\end{pmatrix}.$

1.48 This makes sense: two 90 degree rotations compose to give a 180 degree rotation, which sends every point $\begin{pmatrix}x\\ y\end{pmatrix}$ to its opposite point $\begin{pmatrix}-x\\ -y\end{pmatrix}$ .

Example:

3.10 More generally, if

$R_{\alpha}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix}\,\qquad R_{\beta}=\begin{pmatrix}\cos\beta&% -\sin\beta\\ \sin\beta&\cos\beta\end{pmatrix}$ are two rotations then the composite is

$R_{\alpha}R_{\beta}=\begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix}\begin{pmatrix}\cos\beta&-\sin\beta\\ \sin\beta&\cos\beta\end{pmatrix}$

$=\begin{pmatrix}\cos\alpha\cos\beta-\sin\alpha\sin\beta&-\cos\alpha\sin\beta-% \sin\alpha\cos\beta\\ \sin\alpha\cos\beta+\cos\alpha\sin\beta&-\sin\alpha\sin\beta+\cos\alpha\cos% \beta\end{pmatrix}$

$=\begin{pmatrix}\cos(\alpha+\beta)&-\sin(\alpha+\beta)\\ \sin(\alpha+\beta)&\cos(\alpha+\beta)\end{pmatrix}$

$=R_{\alpha+\beta}.$ (using trigonometric addition formulas). This is what we expect, of course: rotating by

$\beta$ and then

$\alpha$ amounts to rotating by

$\alpha+\beta$ .

Example:

4.25 Let $I=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ be the identity matrix and $M$ be any matrix. Then

$IM=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\begin{pmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{pmatrix}$

$=\begin{pmatrix}M_{11}&M_{12}\\ M_{21}&M_{22}\end{pmatrix}$

$=M.$ Similarly,

$MI=M$ . As you can see, the identity matrix really plays the role of the number

$1$ here.

Example:

6.14 Let $A=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ and $B=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ . Then

$AB=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ but

$BA=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.$ This shows that the order in which we multiply matrices matters:

$AB\neq BA$ . So matrix multiplication is not commutative!

8.40 As an exercise, can you think of a matrix $B$ which does not commute with $A=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ ?