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Question
If possible, using elementary row transformations, find the inverse of the following matrices
`[(2, -1, 3),(-5, 3, 1),(-3, 2, 3)]`
Solution
Here, A = `[(2, -1, 3),(-5, 3, 1),(-3, 2, 3)]` for elementary row transformation
We put A = IA
`[(2, -1, 3),(-5, 3, 1),(-3, 2, 3)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]"A"`
R2 → R2 + R1
`[(2, -1, 3),(-3, 2, 4),(-3, 2, 3)] = [(1, 0, 0),(1, 1, 0),(0, 0, 1)]"A"`
R3 → R3 – R2
`[(2, -1, 3),(-3, 2, 4),(0, 0, -1)] = [(1, 0, 0),(1, 1, 0),(-1, -1, 1)]"A"`
R1 → R1 + R2
`[(-1, 1, 7),(0, -1, -17),(0, 0, -1)] = [(2, 1, 0),(-5, -2, 0),(-1, -1, 1)]"A"`
R1 → R1 + R2 and R3 → –1 . R3
`[(-1, 0, -10),(0, -1, -17),(0, 0, -1)] = [(-3, -1, 0),(-5, -2, 0),(-1, -1, 1)]"A"`
R1 → R1 + 10R3 and R2 → R2 + 17R3
`[(-1, 0, 0),(0, -1, 0),(0, 0, 1)] = [(7, 9, -10),(12, 15, -17),(1, 1, -1)]"A"`
R1 → – 1.R1 and R2 → – 1.R2
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)] = [(-7, 9-, 10),(-12, -15, 17),(1, 1, -1)]"A"`
Hence, A–1 = `[(-7, 9-, 10),(-12, -15, 17),(1, 1, -1)]`
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