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Question
If possible, using elementary row transformations, find the inverse of the following matrices
`[(2, 0, -1),(5, 1, 0),(0, 1, 3)]`
Solution
Here, A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]`
Put A = IA
`[(2, 0, -1),(5, 1, 0),(0, 1, 3)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]"A"`
R1 → 3R1 – R2
`[(1, -1, -3),(5, 1, 0),(0, 1, 3)] = [(3, -1, 0),(0, 1, 0),(0, 0, 1)]"A"`
R2 → R2 – 5R1
`[(1, -1, -3),(0, 6, 15),(0, 1, 3)] = [(3, -1, 0),(-15, 6, 0),(0, 0, 1)]"A"`
R2 → R2 – 5R3
`[(1, -1, -3),(0, 1, 0),(0, 1, 3)] = [(3, -1, 0),(-15, 6, -5),(0, 0, 1)]"A"`
R3 → R3 – R2
`[(1, -1, -3),(0, 1, 0),(0, 0, 3)] = [(3, -1, 0),(-15, 6, -5),(15, -6, 6)]"A"`
R1 → R1 + R2
`[(1, 0, -3),(0, 1, 0),(0, 0, 3)] = [(-12, 5, -5),(-15, 6, -5),(15, -6, 6)]"A"`
`"R"_3 -> 1/3 "R"_3`
`[(1, 0, -3),(0, 1, 0),(0, 0, 1)] = [(-12, 5, -5),(-15, 6, -5),(5, -2, 2)]"A"`
R1 → R1 + 3R3
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)] = [(3, -1, 1),(-15, 6, -5),(5, -2, 2)]"A"`
Hence, `"A"^-1 = [(3, -1, 1),(-15, 6, -5),(5, -2, 2)]`
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