Questions

01

Some matrix products

Compute the following matrix products by hand:

• (a) $\left(\begin{array}{cc}2& 3\\ 1& -5\end{array}\right)\left(\begin{array}{ccc}4& 3& 6\\ 1& -2& 3\end{array}\right)$
• (b) $\left(\begin{array}{ccc}1& 3& -5\\ 0& -2& 1\\ 0& 0& 4\end{array}\right)\left(\begin{array}{ccc}2& 1& 1\\ 0& -1& 3\\ 0& 0& 2\end{array}\right)$

In both (a) and (b) there are three matrices: two are given and their product is the third. In (a), what size vectors do these three matrices act upon? (How many components?) In (b), what do you notice about the nature of the two matrices and their resulting product? Can you guess a rule that generalizes this fact?

02

Double rotation

Show that ${\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)}^2=\left(\begin{array}{cc}\cos (2\theta )& -\sin (2\theta )\\ \sin (2\theta )& \cos (2\theta )\end{array}\right)$, in other words two rotations by $\theta$ is equivalent to one rotation by $2\theta$.

03

Permutation matrices

Explain why the product of two permutation matrices is always a permutation matrix. (Recall the definition: a matrix is a permutation matrix when each row and each column contains exactly one $1$ and the other entries are $0$.)

04

Linearity / Distributivity of matrix actions

Show that $A(\mathbf{u}+\mathbf{v})=A\mathbf{u}+A\mathbf{v}$ whenever a matrix $A$ acts on vectors $\mathbf{u}$ and $\mathbf{v}$.

Hint: first try this with a $3\times 3$ matrix and 3D vector with arbitrary entries. Then to show it in general, use the “summation notation” for the matrix action on a vector:

$$(A\mathbf{u}{)}_i=\sum _{j=1}^n{a}_{ij}{u}_j.$$