SOLVING WORD PROBLEMS USING INVERSE MATRIX

Problem 1 :

If 

find the products AB and BA and hence solve the system of equations x + y + 2z =1,3x + 2y + z = 7,2x + y + 3z = 2.

Solution :

AB  =  BA

x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2

|A|  =  1(6 - 1) - 1(9 - 2) + 2(3 - 4)

  =  1(5) - 1(7) + 2(-1)

  =  5 - 7 - 2

  =  5 - 9

  =  -4

x  =  (-1/4)(-8)  =  2

y  =  (-1/4)(-4)  =  1

y  =  (-1/4)(4)  =  -1

So, the values of x, y and z area 2, 1 and -1 respectively.

Problem 2 :

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" and "y" be the monthly salary and a fixed amount of annual increment respectively.

x + 3y  =  19800 ---------(1)

x + 9y  =  23400  ---------(2)

X = (1/|A|) adj A

|A|  =  9 - 3  =  6

x  =  (178200 - 70200)/6  =  18000

y  =  (-19800 + 23400)/6  =  600

So, the monthly salary is 18000 and annual increment is 600.

Problem 3 :

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 "x" be the number of days taken by men and "y" be the number of days taken by women.

One day work done by 1 men  =  1/x

One day work done by 1 women  =  1/y

(4/x) + (4/y)  =  (1/3)

(2/x) + (5/y)  =  (1/4)

1/x  =  a, 1/y  =  b

4a + 4b  =  1/3  ---(1)

2a + 5b  =  1/4  ---(2)

|A|  =  20 - 8  =  12

x  =  18 and y = 36

So, men can take 18 days to finish the work and women can take 36 days to finish the work.

Problem 4 :

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 :

2x + 5z - 4y  =  15000

3x + y - 2z  =  1000

3y + z - x  =  4000

2x - 4y + 5z  =  15000  --------(1)

3x + y - 2z  =  1000  --------(2)

-x + 3y + z  =  4000  --------(3)

|A|  =  2(1 + 6) + 4(3.- 2) + 5(9 + 1)

  =  2(7) + 4(1) + 5(10)

  =  14 + 4 + 50

|A|  =  68

x  =   (1/68) (136000)  =  2000

y  =   (1/68) (68000)  =  1000

z  =   (1/68) (204000)  =  3000

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.