#### Question

Given two matrices A and B

`A = [(1,-2,3),(1,4,1),(1,-3, 2)] and B = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

#### Solution

`AB = [(1,-2,3),(1,4,1),(1,-3,2)][(11,-5,-14),(-1,-1,2),(-7,1,6)]`

`AB = [(11+3-21,-5+2+3,-14-4+18),(11-4-7,-5-4+1, -14+8+6),(11+3-14, -5+3+2,-14-6+12)]`

`AB = [(-8,0,0),(0,-8,0),(0,0,-8)] = - 8I`

`-1/8 AB = I`

`A(-1/8 B) = I`

`A^(-1) = -1/8 B`

Let AX= C

`[(1,-2,3),(1,4,1),(1,-3,2)][(x),(y),(z)] = [(6),(12),(1)]`

AX = C

`X= A^(-1)C`

we know that `A^(-1) = (-1)/8 B`

`[(x),(y),(z)] = (-1)/8 [(11,-5,-14),(-1,-1,2),(-7,1,6)] [(6),(12),(1)]`

`[(x),(y),(z)] = (-1)/8[(66,-60,-14),(-6,-12,+2),(-42,+12,+6)]`

`[(x),(y),(z)] =(-1)/8 [(8),(-16),(-24)]`

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

x = 1

y = 2

z = 3