Find the following matrices:a) A + B.(b) A - B.(c) -4A.A=begin{bmatrix}2 & -10&-2 14 & 12&104&-2&2 end{bmatrix} , B=begin{bmatrix}6 & 10&-2 0 & -12&-4-5&2&-2 end{bmatrix}

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.

The reduced row echelon form of the matrix A is
$\left[\begin{array}{ccc}1& 2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$
Find three different such matrices A. Explain how you determined your matrices.

$\left(\begin{array}{cc}1& 2\\ 3& -2\end{array}\right)+\left(\begin{array}{cc}-6& -5\\ -2& -2\end{array}\right)=$ ?

What is $\left(\begin{array}{cc}-3& 2\\ -6& 5\end{array}\right)$ plus $\left(\begin{array}{cc}6& -3\\ -6& 2\end{array}\right)$ ?

verify that $(AB{)}^T=B^T{A}^T$
If $A=\left[\begin{array}{cc}2& 1\\ 6& 3\\ -2& 4\end{array}\right]\text{ and }B=\left[\begin{array}{cc}2& 4\\ 1& 6\end{array}\right]$

One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about line. Identify both and briefly justify your choice.