Prove: If A and B are n times n diagonal matrices, then AB = BA.

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Add the following matrices. $\left[\begin{array}{cc}-4& -4\\ 5& -1\\ -3& 1\end{array}\right],\left[\begin{array}{cc}0& 0\\ 3& 1\\ -4& 1\end{array}\right]$

(square roots of the identity matrix) For how many $2\times 2$ matrices A is it true that $A^2=I$ ? Now answer the same question for $n\times x$ n matrices where $n>2$

Given:
$A=[[-2,3],[0,1]]$
$B=[[8,1],[5,4]]$
Find the following matrices:
a. A + B
b. A - B
c. -4A
d. 3A + 2B.

Use the definition of the matrix exponential to compute eA for each of the following matrices:
$A=\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Given A = {1, 2, 3, 4} and B = {x, y, z}, Let R be the following relation from A to B: R = {(1, y),(1, z),(3, y),(4, x),(4, z)} A. Determine the matrix of the relation.