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Question
Find the matrix of the co-factor for the following matrix.
`[(1, 0, 2),(-2, 1, 3),(0, 3, -5)]`
Solution
Let A = `[(1,0,2),(-2,1,3),(0,3,-5)]`
Here, a11 = 1
∴ M11 = `|(1,3),(3,-5)|` = −5 − 9 = −14
and A11 = (−1)1+1 (−14) = −14
a12 = 0
∴ M12 = `|(-2,3),(0,-5)|` = 10 − 0 = 10
and A12 = (−1)1+2 (10) = −10
a13 = 2
∴ M13 = `|(-2,1),(0,3)|` = −6 − 0 = −6
and A13 = (−1)1+3 (−6) = −6
a21 = −2
∴ M21 = `|(0,2),(3,-5)|` = 0 − 6 = −6
and A21 = (−1)2+1 (−6) = 6
a22 = 1
∴ M22 = `|(1,2),(0,-5)|` = −5 − 0 = −5
and A22 = (−1)2+2 (−5) = −5
a23 = 3
∴ M23 = `|(1,0),(0,3)|` = 3 − 0 = 3
and A23 = (−1)2+3 (3) = −3
a31 = 0
∴ M31 = `|(0,2),(1,3)|` = 0 − 2 = −2
and A31 = (−1)3+1 (−2) = −2
a32 = 3
∴ M32 = `|(1,2),(-2,3)|` = 3 + 4 = 7
and A32 = (−1)3+2 (7) = −7
a33 = −5
∴ M33 = `|(1,0),(-2,1)|` = 1 − 0 = 1
and A33 = (−1)3+3 (1) = 1
∴ The matrix of the co-factor is
`[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]` = `[(-14,-10,-6),(6,-5,-3),(-2,-7,1)]`
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