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Question
Find the inverse of the following matrix (if they exist):
`[(2,-3),(5,7)]`
Solution
Let A = `[(2,-3),(5,7)]`
∴ |A| = `|(2,-3),(5,7)| = 14 + 15 = 29 ne 0`
∴ A-1 exists.
Consider AA-1 = I
∴ `[(2,-3),(5,7)] "A"^-1 = [(1,0),(0,1)]`
By 3R1, we get,
`[(6,-9),(5,7)] "A"^-1 = [(3,0),(0,1)]`
By R1 - R2, we get,
`[(1,-16),(5,7)] "A"^-1 = [(3,-1),(0,1)]`
By R2 - 5R1, we get,
`[(1,-16),(0,87)] "A"^-1 = [(3,-1),(-15,6)]`
By `(1/87)"R"_2,`we get
`[(1,-16),(0,1)] "A"^-1 = [(3,-1),(-5/29,2/29)]`
By R1 + 16R2, we get,
`[(1,0),(0,1)]"A"^-1 = [(7/29,3/29),(-5/29,2/29)]`
∴ A-1 = `1/29 [(7,3),(-5,2)]`
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