Application of Matrices

Express the following equations in matrix form and solve them by the method of reduction:

x − y + z = 1, 2x − y = 1, 3x + 3y − 4z = 2

Complete the following activity.

The cost of 4 kg potato, 3kg wheat and 2kg rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3kg rice is ₹ 45. The cost of 6 kg potato, 3 kg rice and 2 kg wheat is ₹ 70. Find the per kg cost of each item by matrix method.

Solution: Let the cost of potato, wheat and rice per kg be x, y and z respectively.

Therefore by given conditions,

4x + ( )y + 2( ) = ( )

x + 2y + ( )( ) = ( )

( )x + 2y + 3z = ( )

Matrix form of above equations is,

`[("( )", 3, "( )"),(1, "( )", 3),("( )", 2, "( )")] [(x),(y),(z)] =[("( )"), (45), ("( )")]`

R₁ ↔ R₂

`[(1, 2, 3),("( )", "( )", "( )"),(6, 2, 3)] [(x),(y),(z)] =[("( )"), (60), ("( )")]`

R₂ – 4R₁, R₃ – 6R₁

`[(1, 2, 3),("( )", -5, "( )"),(0, "( )", -15)] [(x),(y),(z)] =[(45), ("( )"), (-200)]`

`(-1)/5 "R"_2, (-1)/5 "R"_3`

`[("( )", 2, 3),(0, "( )", 2),(0, 2, "( )")] [(x),("( )"),(z)] =[(45), (24), (40)]`

R₃ – 2R₂

`[(1, 2, 3),(0, 1, 2),(0, 0, -1)] [(x),(y),(z)] =[("( )"), ("( )"), ("( )")]`

By pre multiplying we get,

x + 2y + ( )z = ( )    .....(i)

y + 2z = 24    ......(ii)

–z = ( )      ......(iii)

From (iii), we get, z = ( )

From (ii), we get, y = ( )

From (i), we get, x = ( )

Therefore the cost of Potato, Wheat and Rice per kg are _______, _______ and _______ respectively.

The total cost of 3 T.V. and 2 V.C.R. is ₹ 35,000. The shopkeeper wants profit of ₹1000 per television and ₹ 500 per V.C.R. He can sell 2 T.V. and 1 V.C.R. and get the total revenue as ₹ 21,500. Find the cost price and the selling price of a T.V. and a V.C.R.

The sum of three numbers is 6. If we multiply third number by 3 and add it to the second number we get 11. By adding the first and third number we get a number which is double the second number. Use this information and find a system of linear equations. Find the three numbers using matrices.

Solve the following equation by the method of inversion:

5x − y + 4z = 5, 2x + 3y + 5z = 2 and 5x − 2y + 6z = −1

Solve the following system of equations by the method of inversion.

x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2

Express the following equations in matrix form and solve them by method of reduction.
x + 3y  = 2, 3x + 5y = 4

Find x, y, z, if `{5[(0, 1),(1, 0),(1, 1)] - [(2, 1),(3, - 2),(1, 3)]} [(2),(1)] = [(x - 1),(y + 1),(2z)]`