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Question
Write Minors and Cofactors of the elements of the following determinant:
`|(1,0,4),(3,5,-1),(0,1,2)|`
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Solution
`|(1,0,4),(3,5,-1),(0,1,2)|`
Minors:
M11 = `|(5,-1),(1,2)|`
= 5 × 2 − (−1) × 1
= 10 + 1
= 11
M12 = `|(3,-1),(0,2)|`
= 3 × 2 − (−1) × 0
= 6 + 0
= 6
M13 = `|(3,5),(0,1)|`
= 3 × 1 − 5 × 0
= 3 − 0
= 3
M21 = `|(0,4),(1,2)|`
= 0 × 2 − 1 × 4
= 0 − 4
= −4
M22 = `|(1,4),(0,2)|`
= 1 × 2 − 4 × 0
= 2 − 0
= 2
M23 = `|(1,0),(0,1)|`
= 1 × 1 − 0 × 0
= 1 − 0
= 1
M31 = `|(0,4), (5, -1)|`
= (−1) × 0 − 4 × 5
= 0 − 20
= −20
M32 = `|(1,4),(3,-1)|`
= 1 × (−1) − 3 × 4
= (−1) − 12
= −13
M33 = `|(1,0),(3,5)|`
= 1 × 5 − 0 × 3
= 5 − 0
= 5
Cofactors:
A11 = (−1)1+1 M11
= 11 × 1
= 11
A12 = (−1)1+2 M12
= (−1) × 6
= −6
A13 = (−1)1+3 M13
= 1 × 3
= 3
A21 = (−1)2+1 M21
= (−1) × (−4)
= 4
A22 = (−1)2+2 M22
= 1 × 2
= 2
A23 = (−1)2+3 M23
= (−1) × 1
= −1
A31 = (−1)3+1 M31
= 1 × (−20)
= −20
A32 = (−1)3+2 M32
= (−1) × (−13)
= −13
A33 = (−1)3+3 M33
= 1 × 5
= 5
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