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 added to the sum of second and third numbers we get 4 . Find the numbers using matrices .

If three numbers are added , their sum is 2 . If two times the second

1.	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 added to the sum of second and third numbers we get 4 . Find the numbers using matrices .
Answer» Let , x,y, z be three numbers. Then according to given conditions , we get the following equations : ` x + y + z = 2 ` ` x - 2y + z = 8 ` and ` 3x + y + z = 4 ` Now , putting these equations is matrix method we get , `[(1,1,1),(1, -2, 1),(3,1,1)][(x),(y),(z)] = [(2),(8),(4)] ` i .e. AX = B Now using , ` R_(2) - R_(1) , R_(3) - 3R_(1) ` , we get `[(1,1,1),(0,-3,0),(0,-2,-2)][(x),(y),(z)] = [(2),(6),(-2)] ` ` therefore x + y + z = 2 `...(i) ` - 3y = 6` `rArr y = - 2 ` and ` - 2 (y + z) = - 2 ` `rArr y + z= 1` ` rArr - 2 + z = 1 " "[ because y = - 2] ` ` rArr z = 3 ` Putting the values of y and z in equations (i) we get , ` x - 2 + 3 = 2 ` ` rArr x + 1 = 2 ` ` rArr x = 1 ` ` therefore ` Three numbers are ` x = 1 , y = - 2 and z = 3 ` .