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Question
Find the inverse of the following matrix (if they exist):
`((2,1),(1,-1))`
Solution
Let A = `((2,1),(1,-1))`
∴ |A| = `|(2,1),(1,-1)| = - 2 - 1 = - 3 ≠ 0`
∴ A-1 exists.
Consider AA-1 = I
∴ `((2,1),(1,-1)) "A"^-1 = ((1,0),(0,1))`
By R1 ↔ R2, we get,
`((1,-1),(2,1)) "A"^-1 = ((0,1),(1,0))`
By R2 - 2R1, we get,
`((1,-1),(0,3)) "A"^-1 = ((0,1),(1,-2))`
By `(1/3) "R"_2`, we get,
`((1,-1),(0,1)) "A"^-1 = ((0,1),(1/3,-2/3))`
By R1 + R2, we get,
`((1,0),(0,1)) "A"^-1 = ((1/3,1/3),(1/3,-2/3))`
∴ A-1 = `1/3((1,1),(1,-2))`
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