Given the system of equation below , identify the number of solutions. begin{cases}y=5x-715x-3y=7end{cases}

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.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$A=\left[\begin{array}{ccc}2& 1& 1\\ -1& -1& 4\end{array}\right],B=\left[\begin{array}{cc}0& 2\\ -4& 1\\ 2& -3\end{array}\right],C=\left[\begin{array}{cc}6& -1\\ 3& 0\\ -2& 5\end{array}\right],D=\left[\begin{array}{ccc}2& -3& 4\\ -3& 1& -2\end{array}\right]$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A

This semester you have learnt how to solve simultaneous equations algebraically. Another method of solving these equations is using Cramers

Find three $2\times 2$ idempotent matrices. (Recall that a square matrix A is idempotent when $A^2=A$)

If $A=\left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right],B=\left[\begin{array}{c}2\\ 1\end{array}\right],C=\left[\begin{array}{cc}-7& 1\\ 0& 4\end{array}\right],D=\left[\begin{array}{ccc}3& 2& 1\end{array}\right]\text{ and }E=\left[\begin{array}{ccc}2& 3& 4\\ 1& 2& -1\end{array}\right]$
Find , if possible,
a) A+B , C-A and D-E b)AB, BA , CA , AC , DA , DB , BD , EB , BE and AE c) 7C , -3D and KE

Find the product AB of the two matrices listed below:
$A=\left(\begin{array}{ccc}2& 1& 3\\ -2& 2& 4\\ -1& -3& -4\end{array}\right)$
$B=\left(\begin{array}{cc}-1& -2\\ 1& 2\\ 3& 4\end{array}\right)$