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Question
Find the inverse of the following matrices by transformation method:
`[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`
Solution
Let A = `[(2, 0, −1),(5, 1, 0),(0, 1, 3)]`
∴ |A| = `|(2, 0, -1),(5, 1, 0),(0, 1, 3)|`
= 2(3 – 0) – 0(15 – 0) – 1(5 – 0)
= 6 – 0 – 5
= 1 ≠ 0
∴ A–1 exists.
Consider AA–1 = I
∴ `[(2, 0, –1),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
By R1→3R1,
`[(6, 0, –3),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(3, 0, 0),(0, 1, 0),(0, 0, 1)]`
By R1 → R1 − R2
`[(1, –1, –3),(5, 1, 0),(0, 1, 3)] "A"^-1 = [(3, –1, 0),(0, 1, 0),(0, 0, 1)]`
By R2 → R2 − 5R1
`[(1, –1, –3),(0, 6, 15),(0, 1, 3)] "A"^-1 = [(3, –1, 0),(–15, 6, 0),(0, 0, 1)]`
By R2↔R3
`[(1, –1, –3),(0, 1, 3),(0, 6, 15)] "A"^-1 = [(3, –1, 0),(0, 0, 1),(–15, 6, 0)]`
By R1→R1 + R2 and R3→R3 − 6R2
`[(1, 0, 0),(0, 1, 3),(0, 0, –3)] "A"^-1 = [(3, –1, 1),(0, 0, 1),(–15, 6, –6)]`
By R3 → `(- 1)/3 "R"_3`,
`[(1, 0, 0),(0, 1, 3),(0, 0, 1)] "A"^-1 = [(3, –1, 1),(0, 0, 1),(5, –2, 2)]`
By R2 →R2 – 3R3
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)] "A"^-1 = [(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`
IA–1 = `[(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`
A–1 = `[(3, –1, 1),(–15, 6, –5),(5, –2, 2)]`
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