WORD PROBLEMS ON SOLVING SYSTEM OF LINEAR EQUATIONS INVERSE MATRIX

Problem 1 :

A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was 19,800 per month at the end of the first month after 3 years of service and 23,400 per month at the end of the first month after 9 years of service, find his starting salary and his annual increment. (Use matrix inversion method to solve the problem.)

Solution :

Let x be the fixed amount and y be amount of annual increment.

x + 3y = 19800 ----(1)

x + 9y = 23400 ----(2)

x = 108000/6 ==> 18000

y = 3600/6 ==> 600

So, his starting salary is 18000 and annual increment is 600.

Problem 2 :

Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.

Solution :

Let 1/x be the time taken by one man in one day and 1/y be the time taken by one woman in one day

Finding inverse of a matrix :

Solving for x and y from the values of a and b :

Problem 3 :

The prices of three commodities A B, and C are  x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C . Person Q purchases 2
units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C . In the process, P Q, and R earn 15,000, 1,000 and 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)

Solution :

P ==> 2x-4y+5z = 15000----(1)

Q ==> 3x+y-2z = 1000----(2)

R ==> -x+3y+z = 4000----(3)

Finding adjoint and inverse of matrix :

Solving for x, y and z :

x = 2000, y = 1000 and z = 3000