Question

Given that:  A = `((1,-1,0),(2,3,4),(0,1,2))`  and B = `((2,2,-4),(-4,2,-4),(2,-1,5))`  , find AB. Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7. 

Answer 1

AB = `[(1,-1,0),(2,3,4),(0,1,2)] [(2,2,-4),(-4,2,-4),(2,-1,5)]`

`=[(2+4+0,  2-2-0, -4+4+0),(4-12+8,  4+6-4, -8-12+20),(0-4+4,  0+2-2, 0-4+10)] = [(6,0,0),(0,6,0),(0,0,6)]`

AB = 6I₃ 

A^-1 AB = 6A^-1 I

`1/6`B = A^-1 

Given system of equations are :

x - y = 3 , 2x + 3y + 4z = 17 and y + 2z = 7

Their matrix form is 

AX = C

⇒ X = A^-1 C

`= 1/6 [(6+34-28),(-12+34-28),(6-17+35)]`

`= 1/6 [(12),(-6),(24)] = [(2),(-1),(4)]`

Hence, x = 2 , y= -1 and z = 4