**Solving Matrices Word Problems Using Inversion Method :**

Here we are going to see some example problems of solving matrices word problems using inversion method.

**Question 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

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

**Question 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

Hence the monthly salary is 18000 and annual increment is 600.

**Question 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

a = (1/12) [5/3 - 1] = (1/12) (2/3) a = 1/18 |
b = (1/12) [-2/3 + 1] = (1/12) (1/3) b = 1/36 |

x = 18 and y = 36

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

**Question 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

After having gone through the stuff given above, we hope that the students would have understood, "Solving Matrices Word Problems Using Inversion Method".

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