Using matrix method, solve x + 2y + z = 7, x + 3z = 11 and 2x - 3y =1.

if A =\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

Write the minors and co-factors of the elements of \(\begin{vmatrix}5 & 3 \\-6 & 2\end{vmatrix}\)

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

Find the inverse of \(\begin{bmatrix}-1 & 5 \\-3 & 2 \end{bmatrix}\).

Solve: 2x + 5y = 1 and 3x + 2y = 7 using matrix method.

The data below are about an economy of two industries P and Q. The values are in lakhs of rupees.

Producer	User		Final Demand	Total output
	P	Q
P	16	12	12	40
Q	12	8	4	24

Find the technology matrix and check whether the system is viable as per Hawkins-Simon conditions.

Using co-factors of elements of  second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)