The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
Let the numbers are x, y, z
X + y + z = 2 ……(i)
Also, 2y + (x + z) + 1
X + 2y + z = 1 ……(ii)
Again,
x + z + 5(x) = 6
5x + y + z = 6 …… (iii)
A X = B
|A| = 1(1) – 1(– 4) + 1(– 9)
= 1 + 4 – 9
= – 4
Hence, the unique solution given by x = A – 1B
C11 = (– 1)1 + 1 (2 – 1) = 1
C12 = (– 1)1 + 2 (1 – 5) = 4
C13 = (– 1)1 + 3 (1 – 10) = – 9
C21 = (– 1)2 + 1 (1 – 1) = 0
C22 = (– 1)2 + 2 (1 – 5) = – 4
C23 = (– 1)2 + 3 (1 – 5) = 4
C31 = (– 1)3 + 1 (1 – 2) = – 1
C32 = (– 1)3 + 2 (1 – 1) = 0
C33 = (– 1)3 + 3 (2 – 1) = 1
X = A – 1 B = 
Adj A = 
X = 
X = 
= 
Hence, 