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Question
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
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Solution
Let A = `[(2,1,3),(4,-1,0),(-7,2,1)]`
Then |A| = `|(2,1,3),(4,-1,0),(-7,2,1)|`
= 2(−1 − 0) − 1(4 − 0) + 3(8 − 7)
= −2 − 4 + 3
= −3 ≠ 0
So, A is a non-singular matrix and therefore, it is invertible. Let Cij be cofactor of aij in A. Then the cofactors of elements of A are given by,
C11 = `(-1)^(1+1) |(-1,0), (2,1)|`
= 1 × (−1 − 0)
= −1
C12 = `(-1)^(1+2) |(4,0), (-7,1)|`
= (−1) × (4 + 0)
= −1 × 4
= −4
C13 = `(-1)^(1+3)|(4,-1),(-7,2)|`
= 1 × (8 − 7)
= 1 × 1
= 1
C21 = `(-1)^(2+1) |(1,3), (2,1)|`
= (−1) × (1 − 6)
= (−1) × (−5)
= 5
C22 = `(-1)^(2+2) |(2,3), (-7,1)|`
= 1 × (2 + 21)
= 1 × 23
= 23
C23 = `(-1)^(2+3) |(2,1), (-7,2)|`
= (−1) × (4 + 7)
= (−1) × 11
= −11
C31 = `(-1)^(3+1) |(1,3), (-1,0)|`
= 1 × (0 + 3)
= 1 × 3
= 3
C32 = `(-1)^(3+2) |(2,3), (4,0)|`
= (−1) × (0 − 12)
= (−1) × (−12)
= 12
C33 = `(-1)^(3+3)|(2,1), (4,-1)|`
= 1 × (−2 − 4)
= 1 × (−6)
= −6
∴ adj A = `[(-1,-4,1),(5,23,-11),(3,12,-6)] = [(-1,5,3),(-4,23,12),(1,-11,-6)]`
A−1 = `1/|A|` adj A
= `1/-3 [(-1,5,3),(-4,23,12),(1,-11,-6)]`
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