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प्रश्न
Find the cofactor matrix, of the following matrices: `[(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]`
उत्तर
The co-factor Aij of aij is equal to (– 1)i+j Mij.
Here, a11 = 5
∴ M11 = `|(-2, 1),(1, 1)|` = – 2 – 1 = – 3
and A11 = (– 1)1+1 M11 = (1) (– 3) = – 3
a12 = 8
∴ M12 = `|(-1, 1),(-2, 1)|` = – 1 + 2 = 1
and A12 = (– 1)1+2 M11 = (–1) (1) = – 1
a13 = 7
∴ M13 = `|(-1, -2),(-2, 1)|` = – 1 – 4 = – 5
and A13 = (– 1)1+3 M13 = (1) (– 5) = – 5
a21 = – 1
∴ M21 = `|(8, 7),(1, 1)|` = 8 – 7 = 1
and A21 = (– 1)2+1 M21 = (– 1) (1) = – 1
a22 = – 2
∴ M22 = `|(5, 7),(-2, 1)|` = 5 + 14 = 19
and A22 = (– 1)2+2 M22 = (– 1) (19) = 19
a23 = 1
∴ M23 = `|(5, 8),(-2, 1)|` = 5 + 16 = 21
and A23 = (– 1)2+3 M23 = (– 1) (21) = – 21
a31 = – 2
∴ M31 = `|(8, 7),(-2, 1)|` = 8 + 14 = 22
and A31 = (– 1)3+2 M31 = (1) (22) = 22
a32 = 1
∴ M32 = `|(5, 7),(-1, 1)|` = 5 + 7 = 12
and A32 = (– 1)3+2 M32 = (– 1) (12) = – 12
a33 = 1
∴ M33 = `|(5, 8),(-1, -2)|` = – 10 + 8 = – 2
and A33 = (– 1)3+3 M33 = (1) (– 2) = – 2
∴ 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, -1, -5),(-1, 19, -21),(22, -12, -2)]`.
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