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Question
Solve the following equation by the method of inversion:
5x − y + 4z = 5, 2x + 3y + 5z = 2 and 5x − 2y + 6z = −1
Solution
The given equations can be written in the matrix form as:
`[(5,-1,4),(2,3,5),(5,-2,6)][("x"),("y"),("z")] = [(5),(2),(-1)]`
This is of the form AX = B, where
A = `[(5,-1,4),(2,3,5),(5,-2,6)], "X" = [("x"),("y"),("z")], "B" = [(5),(2),(-1)]`
Let us find A–1.
|A| = `|(5,−1,4),(2,3,5),(5,−2,6)|`
= 5(18 + 10) + 1(12 − 25) + 4(− 4 − 15)
= 140 − 13 − 76
= 51 ≠ 0
∴ A-1 exists.
Now, we have to find the co-factor matrix
`= ["A"_"ij"]_(3xx3), "where" A"ij" = (-1)^("i" + "j")"M"_"ij"`
`"A"_11 = (-1)^(1+1)"M"_11 = |(3,5),(-2,6)| = 18 + 10 = 28`
`"A"_12 = (-1)^(1+2)"M"_12 = - |(2,5),(5,6)| = - (12 - 25) = 13`
`"A"_13 = (-1)^(1+3)"M"_13 = |(2,3),(5,-2)| = - 4 - 15 = - 19`
`"A"_21 = (-1)^(2+1)"M"_21 = -|(-1,4),(-2,6)| = - (- 6 + 8)= - 2`
`"A"_22 = (-1)^(2+2)"M"_22 = |(5,4),(5,6)| = (30 - 20)= 10`
`"A"_23 = (-1)^(2+3)"M"_23 = - |(5,-1),(5,-2)| = - (- 10 + 5)= 5`
`"A"_31 = (-1)^(3+1)"M"_31 = |(-1,4),(3,5)| = - 5 - 12 = - 17`
`"A"_32 = (-1)^(3 + 2)"M"_32 = - |(5,4),(2,5)| = - ( 25 - 8)= - 17`
`"A"_33 = (-1)^(3+3)"M"_33 = |(5,-1),(2,3)| = 15 + 2 = 17`
∴ the cofactor matrix =
`[("A"_11,"A"_12,"A"_13),("A"_21,"A"_22,"A"_23),("A"_31,"A"_32,"A"_33)] = [(28,13,-19),(-2,10,5),(-17,-17,17)]`
∴ adj A = `[(28,-2,-17),(13,10,-17),(-19,5,17)]`
∴ `"A"^-1 = 1/|"A"|`(adj A)
`= 1/51 [(28,-2,-17),(13,10,-17),(-19,5,17)]`
Now, premultiply AX = B by A–1, we get,
`"A"^-1 ("AX") = "A"^-1"B"`
∴ `("A"^-1"A")"X" = "A"^-1"B"`
∴ IX = `"A"^-1"B"`
∴ X = `1/51 [(28,-2,-17),(13,10,-17),(-19,5,17)][(5),(2),(-1)]`
`= 1/51[(140 - 4 + 17),(65 + 20 + 17),(- 95 + 10 - 17)]`
`= 1/51 [(153),(102),(-102)]`
∴ `[("x"),("y"),("z")] = [(3),(2),(-2)]`
By equality of matrices,
x = 3, y = 2, z = −2 is the required solution.
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