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Question
If A = `[(-5, 1, 3),(7, 1, -5),(1, -1, 1)]` and B = `[(1, 1, 2),(3, 2, 1),(2, 1, 3)]`, Find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2
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Solution
A = `[(-5, 1, 3),(7, 1, -5),(1, -1, 1)]`
B = `[(1, 1, 2),(3, 2, 1),(2, 1, 3)]`
AB = `[(-5, 1, 3),(7, 1, -5),(1, -1, 1)] [(1, 1, 2),(3, 2, 1),(2, 1, 3)]`
= `[(- 5 + 3 + 6, - 5 + 2 + 3, -10 + 1 + 9),(7 + 3 -10, 7 + 2 - 5, 14 + 1 - 15),(1 - 3 + 2, 1 - 2 + 1, 2 - 1 + 3)]`
= `[(4, 0, 0),(0, 4, 0),(0, 0, 4)]`
= 4I3
BA = `[(1, 1, 2),(3, 2, 1),(2, 1, 3)][(-5, 1, 3),(7, 1, -5),(1, -1, 1)]`
= `[(-5 + 7 + 2, 1 + 1 - 2, 3 - 5 + 2),(-15 + 14 + 1, 3 + 2 + -1, 9 - 10 + 1),(-10 + 7 + 3, 2 + 1 - 3, 6 - 5 + 3)]`
= `[(4, 0, 0),(0, 4, 0),(0, 0, 4)]`
= 4I3
AB = BA = 4I3
`(1/4 "A")"B" = "B"(1/4 "A")` = I
⇒ `"B"^-"I" = 1/4"A"`
Matrix form `[(1, 1, 2),(3, 2, 1),(2, 1, 3)][(x),(y),(z)] = [(1),(7),(2)]`
BX = C
X = `"B"^-1"C"`
`[(x),(y),(z)] = 1/4"A"[(1),(7),(2)]`
= `1/4[(-5, 1, 3),(7, 1, -5),(1, -1, 1)][(1),(7),(2)]`
= `1/4[(- 5 + 7 + 6),(7 + 7 - 10),(1 - 7 + 2)]`
= `1/4[(8),(4),(-4)]`
= `[(2),(1),(-1)]`
∴ x = 2, y = 1, z = – 1
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