Determine whether each statement is sometimes, always , or never true for matrices A and B and explain your reasoning. Stuck on one of them? Try a few examples. a) If A+B exists , then A-B exists b) If A and B have the same number of elements , then A+B exists.

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.

Define Exponential Matrices?

Let $A=I_2$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ . Discuss the validity of the following statement and cite reasons for your conclusion:
If A and B have the same trace , determinant , rank , and eigenvalues , then the matrices are similar.

How these pair are taken?What is the rule behind it?

Explain why the formula is not valid for matrices. Illustrate your argument with examples.
$(A+B)(A+B)=A^2+2AB+B^2$

Perform the indicated matrix operations:
$\left[\begin{array}{cc}5& 4\\ -3& 7\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}-4& 8\\ 6& 0\\ -5& 3\end{array}\right]$