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Question
Find the inverse of the following matrix (if they exist):
`[(2,0,-1),(5,1,0),(0,1,3)]`
Solution
Let A = `[(2,0,-1),(5,1,0),(0,1,3)]`
∴ |A| = `|(2,0,-1),(5,1,0),(0,1,3)|`
= 2(3 - 0) - 0 - 1(5 - 0)
= 6 - 0 - 5
= 1 ≠ 0
∴ A-1 exists.
Consider AA-1 = I
∴ `[(2,0,-1),(5,1,0),(0,1,3)] "A"^-1 = [(1,0,0),(0,1,0),(0,0,1)]`
By R1 ↔ R2, we get,
`[(5,1,0),(2,0,-1),(0,1,3)] "A"^-1 = [(0,1,0),(1,0,0),(0,0,1)]`
By R1 - 2R2, we get,
`[(1,1,2),(2,0,-1),(0,1,3)] "A"^-1 = [(-2,1,0),(1,0,0),(0,0,1)]`
By R2 - 2R1, we get,
`[(1,1,2),(0,-2,-5),(0,1,3)] "A"^-1 = [(-2,1,0),(5,-2,0),(0,0,1)]`
By R2 + 3R3, we get,
`[(1,1,2),(0,1,4),(0,1,3)] "A"^-1 = [(-2,1,0),(5,-2,3),(0,0,1)]`
By R1 - R2 and R3 - R2, we get,
`[(1,0,-2),(0,1,4),(0,0,-1)] "A"^-1 = [(-7,3,-3),(5,-2,3),(-5,2,-2)]`
By (- 1)R3, we get,
`[(1,0,-2),(0,1,4),(0,0,1)] "A"^-1 = [(-7,3,-3),(5,-2,3),(5,-2,2)]`
By `"R"_1 + 2"R"_3` and `"R"_2 - 4"R"_3`, we get,
`[(1,0,0),(0,1,0),(0,0,1)] "A"^-1 = [(3,-1,1),(-15,6,-5),(5,-2,2)]`
`therefore "A"^-1 = [(3,-1,1),(-15,6,-5),(5,-2,2)]`
Notes
The answer in the textbook is incorrect.
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