Given A = {1, 2, 3, 4} and B = {x, y, z}, Let R be the following relation from A to B: R = {(1, y),(1, z),(3, y),(4, x),(4, z)} A. Determine the matrix of the relation.

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

Solve the simultaneous Linear equations using matrix inverse Method.
8x+3y=2
6x+2y=4

Find a family of the matrices that is similar to the matrix $Q=\left[\begin{array}{cc}p& -2\\ 4& -3\end{array}\right]$

Let W be the subspace of all diagonal matrices in $M_{2,2}$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

For Exercise , perform the indicated operations if possible.
$A=\left[\begin{array}{ccc}4& 1& -3\\ 2& 4& 6\end{array}\right],B=\left[\begin{array}{cc}1& 9\\ 0& -1\\ 3& 5\end{array}\right],C=\left[\begin{array}{ccc}0& 1& -4\\ 2& -1& 8\end{array}\right]$
A+B=?

For the matrices (a) find k such that Nul A is a subspace of ${\displaystyle {\mathbb{R}}^k}$, and (b) find k such that Col A is a subspace of ${\displaystyle {\mathbb{R}}^k}$
$$A=\left[\begin{array}{ccccc}4& 5& -2& 6& 0\\ 1& 1& 0& 1& 0\end{array}\right]$$