If A = \(\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix},\) then |2A| is equal to ________.

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by ________.

If \(\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0\) then the value of x is ________.

If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is ________.

If any three rows or columns of a determinant are identical then the value of the determinant is ________.

Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.

Find adj A for \(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)

If \(A=\left| \begin{matrix} -2 & 6 \\ 3 & -9 \end{matrix} \right| \)then, find A^-1

Evaluate\(\left| \begin{matrix} 1 & a & { a }^{ 2 } \\ 1 & b & { b }^{ 2 } \\ 1 & c & { c }^{ 2 } \end{matrix} \right| \)= (a–b) (b–c) (c–a)

If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A ^-1.

The technology matrix of an economic system of two industries is \(\left[ \begin{matrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{matrix} \right] \) Test whether the system is viable as per Hawkins – Simon conditions.

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs.320. The cost of 2kg onion, 4 kg wheat and 6 kg rice is Rs.560. The cost of 6 kg onion, 2 kg wheat and 3 kg rice is Rs. 380. Find the cost of each item per kg by matrix inversion method.

An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other’s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that the system is viable? 2 and 5 labour days are required to produce a tonnes of coal and steel respectively. If economy needs 100 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required.