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Question
Find inverse, by elementary row operations (if possible), of the following matrices
`[(1, 3),(-5, 7)]`
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Solution
Let A = `[(1, 3),(-5, 7)]`
|A| = 1 × 7 – (– 5) × 3
= 7 + 15
= 22 ≠ 0
So, A is invertible.
Let A = IA
⇒ `[(1, 3),(-5, 7)] = [(1, 0),(0, 1)]"A"`
R2 → R2 + 5R1
⇒ `[(1, 3),(0, 22)] = [(1, 0),(5, 1)]"A"`
`"R"_2 -> 1/22 "R"_2`
⇒ `[(1, 3),(0, 1)] = [(1, 0),(5/22, 1/22)]"A"`
R1 → R1 + 3R2
⇒ `[(1, 0),(0, 1)] = [(7/22, (-3)/22),(5/22, 1/22)]"A"`
So `"A"^-1 = [(7/22, (-3)/22),(5/22, 1/22)]`
⇒ `1/22[(7, -3),(5, 1)]`
Hence, inverse of `[(1, 3),(-5, 7)]` is `1/22 [(7, -3),(5, 1)]`
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