Question

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

Answer 1

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

X = A^-1 . B   ......(4)

Now

|A| = `|(1,1,1),(0,-3,0),(0,-2,-2)| [(x),(y),(z)] = [(2),(6),(-2)]`

`|(x+,y+,z),(0 - ,3y ,+ 0),(0,-2,-2)|  = [(2),(6),(-2)]`

By equality of matrices,

x + y + z = 2 ……(1)

-3y = 6 ……(2)

– 2y – 2z = -2 ……..(3)

From (2), y = -2

Substituting y = -2 in (3), we get,

-2(-2) – 2z = -2

∴ -2z = -6

∴ z = 3

Substituting y = -2, z = 3 in (1), we get,

x – 2 + 3 = 2

∴ x = 1

Hence, the required numbers are 1, -2 and 3.