Evaluate:\(\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}\)

Prove that \(\begin{vmatrix} -a^{ 2 } & ab & ac \\ ab & -b^{ 2 } & bc \\ ac & bc & -c^{ 2 } \end{vmatrix}=4a^{ 2 }b^{ 2 }{ c }^{ 2 }\)

If A = \(\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 7\\1 & -1 & 1 \end{bmatrix}\) verify that A ( adj A ) = ( adj A ) A = |A| I₃.

Suppose the inter-industry flow of the product of two industries are given as under.

Production sector	Consumption sector		Domestic demand	Total output
	X	Y
X	30	40	50	120
Y	20	10	30	60

Determine the technology matrix and test Hawkin's -Simon conditions for the viability of the system. If the domestic demand changes to 80 and 40 units respectively, what should be the gross output of each sector in order to meet the new demands.

The cost of 2 Kg of Wheat and 1 Kg of Sugar is Rs.70. The cost of 1 Kg of Wheat and 1 Kg of Rice is Rs.70 The cost of 3 Kg of Wheat, 2 Kg of Sugar and 1 Kg of rice is Rs.170. Find the cost of per kg each item using matrix inversion method.