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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
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