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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.

Determine which matrices are in reduced echelon form and which others are only in echelon form.
a)$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
b)$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
c)$\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$ ф

Determine the null space of each of the following matrices:
$\left[\begin{array}{cccc}1& 1& -1& 2\\ 2& 2& -3& 1\\ -1& -1& 0& -5\end{array}\right]$

Which of the following sets are equal?
A = {a,b,c,d}
B = {d,e,a,c}
C = {d,b,a,c}
D = {a,a,d,e,c,e}

For the given systems of linear equations, determine the values of $b_1,b_2,\text{ and }{b}_3$ necessary for the system to be consistent. (Using matrices)
$x-y+3z=b_1$
$3x-3y+9z=b_2$
$-2x+2y-6z=b_3$

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 ?