Find the inverse of the transformation x'= 2x - 3y, y' = x + y, that is, find x, y in terms of x' , y' . (Hint: Use matrices.) Is the transformation orthogonal?

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

Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1),(1, 1),(1, 2),(2, 0),(2, 2),(3, 0). Find reflexive, symmetric and transitive closure of R.

The temperature of coffee sold at the Coffee Bean Cafe follows the normal probability distribution, with a mean of 150 degrees. The standard deviation of this distribution is 5 degrees. (a) What is the probability that the coffee temperature is between 150 degrees and 154 degrees? (b) What is the probability that the coffee temperature is more than 164 degrees?

Find the matrices:
a)A + B
b) A - B
c) -4A
d)3A + 2B
$A=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}5& 9\\ 0& 7\end{array}\right]$

If $A=\left[\begin{array}{ccc}1& 2& 0\\ 1& 1& 0\\ 1& 4& 0\end{array}\right],B=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& -1\\ 2& 2& 2\end{array}\right]\text{ and }C=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& -1\\ 1& 1& 1\end{array}\right]$. Show that $AB=AC\text{ but }B\ne C$

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
${\displaystyle 2x_1-5x_2+8x_3=0}$
${\displaystyle -2x_1-7x_2+x_3=0}$
${\displaystyle 4x_1+2x_2+7x_3=0}$