If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices.

#### Solution

Let the three numbers x , y , z.

From given condition, we have

x + y + z = 2 .......(1)

x + z - 2y = 8

x - 2y + z = 8 ......(2)

And

3x + y + z = 4 .....(3)

Given all equation can be written in matrix form as ,

`[(1,1,1),(1,-2,1),(3,1,1)] [(x),(y),(z)] = [(2),(8),(4)]`

Consider , AX = B

On multiplying A^{-1} both sides , we get

A^{-1} . A.X = A^{-1}.B

X = A^{-1} . B ......(4)

Now

A^{-1} = `1/|"A"| "Adj (A)"`

|A| = `|(1,1,1),(1,-2,1),(3,1,1)|`

|A| = 6

Now , For Adj (A) , we need minors and co-factors.

M_{11} = -3 , M_{12} = -2 , M_{13} = 7

M_{21} = 0 , M_{22} = -2 , M_{23} = -2

M_{31} = 3 , M_{32} = 0 ,M_{33} = -3

Therefore ,

A^{-1} = `1/6 [(-3,0,3),(2,-2,0),(7,2,-3)]`

From equation (4),

X = `1/6 [(-3,0,3),(2,-2,0),(7,2,-3)] [(2),(8),(4)]`

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

Hence , x = 1 , y = -2 , z = 3