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Question
Find the adjoint of the following matrices : `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
Solution
Let A = `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
Here,
a11 = 1
∴ M11 = `|(3, 5),(0, -1)|` = – 3 – 0 = – 3
and A11 = (– 1)1+1 (– 3) = – 3
a12 = – 1
∴ M12 = `|(-2, 5),(-2, -1)|` = 2 + 10 = 12
and A12 = (– 1)1+2 (12) = – 12
a13 = 2
∴ M13 = `|(-2, 3),(-2, 0)|` = 0 + 6 = 6
and A13 = (– 1)1+3 (6) = 6
a21 = – 2
∴ M21 = `|(-1, 2),(0, -1)|` = 1 – 0 = 1
and A21 = (– 1)2+1 (1) = – 1
a22 = 3
∴ M22 = `|(1, 2),(-2, -1)|` = – 1 + 4 = 3
and A22 = (– 1)2+2 (3) = 3
a23 = 5
∴ M23 = `|(1, -1),(-2, 0)|` = 0 – 2 = – 2
and A23 = (– 1)2+3 (– 2) = 2
a31 = – 2
∴ M31 = `|(-1, 2),(3, 5)|` = –5 – 6 = – 11
and A31 = (– 1)3+1 (– 11) = – 11
a32 = 0
∴ M32 = `|(1, 2),(-2, 5)|` = 5 + 4 = 9
and A32 = (– 1)3+2 (9) = – 9
a33 = – 1
∴ M33 = `|(1, -1),(-2, 3)|` = 3 – 2 = 1
and A33 = (– 1)3+3 (1) = 1
∴ The matrix of the co-factors is
[Aij]3x3 = `[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]`
= `[(3, -12, 6),(-1, 3, 2),(-11, -9, 1)]`
Now, adj A = `["A"_"ij"]_(3xx3)^"T" = [(-3, -1, -11),(-12, 3, -9),(6, 2, 1)]`.
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