Given the matrices A=begin{bmatrix}5 & 3 -3 & -1 -2 & -5 end{bmatrix} text{ and } B=begin{bmatrix}0 & -2 1 & 3 4 & -3 end{bmatrix} find the 3x2 matrix X that is a solution of the equation. 2X-A=X+B X=?

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.

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

$\left(\begin{array}{cc}6& 7\\ -4& -6\end{array}\right)+\left(\begin{array}{cc}4& 5\\ -2& -5\end{array}\right)=$ ?

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.

Solve the system of linear equations using matrices.
$3x-2y-4=0$
$2y=12-x$

Define is subtraction of matrices commutative and associative?

Determine if b is a linear combination of the vectors formed from the columns of the matrix A.
$A=\left[\begin{array}{ccc}1& -4& 2\\ 0& 3& 5\\ -2& 8& -4\end{array}\right],b=\left[\begin{array}{c}3\\ -7\\ -3\end{array}\right]$