If A,B are symmetric matrices, then prove that (BA^{-1})^T(A^{-1}B^T)^{-1} = I

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.

What is a unit matrix?

Given the system of equation below , identify the number of solutions.
$\{\begin{array}{l}y=5x-7\\ 15x-3y=7\end{array}$

Let $A=\left[\begin{array}{ccc}1& 2& -1\\ 1& 0& 3\\ 1& 1& -1\end{array}\right]$
i) Find the determinant of A
ii)Verify that the matrix
$\left[\begin{array}{ccc}-\frac{3}{4}& \frac{1}{4}& \frac{6}{4}\\ 1& 0& -1\\ \frac{1}{4}& \frac{1}{4}& -\frac{1}{2}\end{array}\right]$ is the inverse matrix of A

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 inverse of the matrix using elementary matrices. $[[2,0],[1,1]]$