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Question
Find the inverse of the matrices (if it exists).
`[(1,2,3),(0,2,4),(0,0,5)]`
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Solution
Let A = `[(1,2,3),(0,2,4),(0,0,5)]`
|A| = `|(1,2,3),(0,2,4),(0,0,5)|`
= 1(10 − 0) − 2(0 − 0) + 3(0 − 0)
= 10 ≠ 0
Now for cofactor Cij = (−1)i+j Mij
C11 = `(-1)^(1+1) |(2,4), (0,5)|` = 10
C12 = `(-1)^(1+2) |(0,4), (0,5)|` = 0
C13 = `(-1)^(1+3)|(0,2),(0,0)|` = 0
C21 = `(-1)^(2+1) |(2,3), (0,5)|` = −10
C22 = `(-1)^(2+2) |(1,3), (0,5)|` = 5
C23 = `(-1)^(2+3) |(1,2), (0,0)|` = 0
C31 = `(-1)^(3+1) |(2,3), (2,4)|` = 2
C32 = `(-1)^(3+2) |(1,3), (0,4)|` = −4
C33 = `(-1)^(3+3)|(1,2), (0,2)|` = 2
∴ Adj A = `[(10,0,0),(-10,5,0),(2,-4,2)]^' = [(10,-10,2),(0,5,-4),(0,0,2)]`
Hence, A−1 = `1/|A|` Adj A
= `1/10 [(10,-10,2),(0,5,-4),(0,0,2)]`
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