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Question
Find the inverse of the following matrix.
`[(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, we get,
`[(6,0,-3),(5,1,0),(0,1,3)]"A"^(−1) = [(3,0,0),(0,1,0),(0,0,1)]`
By R1 → R1 − R2, We get,
`[(1,-1,-3),(5,1,0),(0,1,3)]"A"^(−1) = [(3,-1,0),(0,1,0),(0,0,1)]`
By R2 → R2 − 5R1, We get,
`[(1,-1,-3),(0,6,15),(0,1,3)]"A"^(−1) = [(3,-1,0),(- 15,6,0),(0,0,1)]`
By R2 → R2 − 5R3, We get,
`[(1,-1,-3),(0,1,0),(0,1,3)]"A"^(−1) = [(3,-1,0),(- 15,6,-5),(0,0,1)]`
By R1→R1 + R2 and R3 →R3 − R2, we get,
`[(1,0,-3),(0,1,0),(0,0,3)]"A"^(−1) = [(-12,5,-5),(- 15,6,-5),(15,-6,6)]`
By R3 →`(1/3)`R3, We got,
`[(1,0,-3),(0,1,0),(0,0,1)]"A"^(−1) = [(-12,5,-5),(- 15,6,-5),(5,-2,2)]`
By R1 →R1 + 3R3, we get,
`[(1,0,0),(0,1,0),(0,0,1)]"A"^(−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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