Given n>1 , show that the set of all 1 times n matrices of positive integers in countable.

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.

Suppose that $\left[\begin{array}{cccc}4& A& 2& 4\\ -7& -4& -4& B\end{array}\right]+\left[\begin{array}{cccc}0& 7& C& 2\\ 3& D& 4& -7\end{array}\right]=\left[\begin{array}{cccc}4& 3& 10& 6\\ -4& -5& 0& -4\end{array}\right]$
What are the values of A,B,C and D ?

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

True or False
If A and B are $n\times n$ lower triangular matrices, then AB is also lower triangular

write B as a linear combination of the other matrices, if possible.
$B=[[2,-2,3],[0,0,-2],[0,0,2]]$
$A_1=[[1,0,0],[0,1,0],[0,0,1]]$
$A_2=[[0,1,1],[0,0,1],[0,0,0]]$
$A_3=[[-1,0,-1],[0,1,0],[0,0,-1]]$
$A_4=[[1,-1,1],[0,-1,-1],[0,0,1]]$

Find the characteristic polynomial of the matrices
$\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]$