If $A = \left|\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right|, find A^{-1}.$ Use it solve the system of equations.

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y- 2z =-3

192.

Using elementary row transformations, find the inverse of the matrix $\mathrm{A} = \left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$

Find the value of x and y if: $2\left[\begin{array}{cc}1& 3\\ 0& \mathrm{x}\end{array}\right] + \left[\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right] = \left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right]$

Let $\mathrm{A}= \left|\begin{array}{ccc}3& 2& 5\\ 4& 1& 3\\ 0& 6& 7\end{array}\right|.$ Express A as sum of two matrices such that one is symmetric and the other is skew symmetric.

If A = $\left[\begin{array}{ccc}1 & 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, verify that A² – 4A – 5I = 0.

If A is an invertible matrix of order 3 and |A| = 5, then find |adj. A|.

If matrix A = (1, 2, 3), write AA’, where A’ is the transpose of matrix A.

Using matrices, solve the following system of equations:
2x – 3y + 5 = 11
3x + 2y – 4z = -5
x + y – 2z= -3

If A = $\left[ \begin{array}{cc}\mathrm{cos\alpha }& - \mathrm{sin\alpha }\\ \mathrm{sin\alpha }& \mathrm{cos\alpha }\end{array} \right]$, then for what value of α is A an identity matrix?

If $\left[ \begin{array}{cc}1 & 2\\ 3 & 4\end{array} \right] \left[ \begin{array}{cc}3 & 1\\ 2 & 5\end{array} \right] = \left[ \begin{array}{cc}7& 11\\ \mathrm{k}& 23\end{array} \right]$, then write the value of k.