Question

Find the cofactor matrix, of the following matrices: ${\displaystyle \left[\begin{array}{ccc}5& 8& 7\\ -1& -2& 1\\ -2& 1& 1\end{array}\right]}$

Answer 1

The co-factor Aij of aij is equal to (– 1)i+j Mij.
Here, a₁₁ = 5

∴ M₁₁ = ${\displaystyle \left|\begin{array}{cc}-2& 1\\ 1& 1\end{array}\right|}$ = – 2 – 1 = – 3
and A₁₁ = (– 1)¹⁺¹ M₁₁ = (1) (– 3) = – 3
a₁₂ = 8

∴ M₁₂ = ${\displaystyle \left|\begin{array}{cc}-1& 1\\ -2& 1\end{array}\right|}$ = – 1  + 2 = 1
and A₁₂ = (– 1)¹⁺² M₁₁ = (–1) (1) = – 1
a₁₃ = 7

∴ M₁₃ = ${\displaystyle \left|\begin{array}{cc}-1& -2\\ -2& 1\end{array}\right|}$ = – 1  – 4 = – 5
and A₁₃ = (– 1)¹⁺³ M₁₃ = (1) (– 5) = – 5
a₂₁ = – 1

∴ M₂₁ = ${\displaystyle \left|\begin{array}{cc}8& 7\\ 1& 1\end{array}\right|}$ = 8  – 7 = 1
and A₂₁ = (– 1)²⁺¹ M₂₁ = (– 1) (1) = – 1
a₂₂ = – 2

∴ M₂₂ = ${\displaystyle \left|\begin{array}{cc}5& 7\\ -2& 1\end{array}\right|}$ = 5  + 14 = 19
and A₂₂ = (– 1)²⁺² M₂₂ = (– 1) (19) = 19
a₂₃ = 1

∴ M₂₃ = ${\displaystyle \left|\begin{array}{cc}5& 8\\ -2& 1\end{array}\right|}$ = 5  + 16 = 21
and A₂₃ = (– 1)²⁺³ M₂₃ = (– 1) (21) = – 21
a₃₁ = – 2

∴ M₃₁ = ${\displaystyle \left|\begin{array}{cc}8& 7\\ -2& 1\end{array}\right|}$ = 8  + 14 = 22
and A₃₁ = (– 1)³⁺² M₃₁ = (1) (22) = 22
a₃₂ = 1

∴ M₃₂ = ${\displaystyle \left|\begin{array}{cc}5& 7\\ -1& 1\end{array}\right|}$ = 5  + 7 = 12
and A₃₂ = (– 1)³⁺² M₃₂ = (– 1) (12) = – 12
a₃₃ = 1

∴ M₃₃ = ${\displaystyle \left|\begin{array}{cc}5& 8\\ -1& -2\end{array}\right|}$ = – 10 + 8 = – 2
and A₃₃ = (– 1)³⁺³ M₃₃ = (1) (– 2) = – 2

∴ The matrix of the co-factors is

[A_ij]_3x3 = ${\displaystyle \left[\begin{array}{ccc}{\text{A}}_{11}& {\text{A}}_{12}& {\text{A}}_{13}\\ {\text{A}}_{21}& {\text{A}}_{22}& {\text{A}}_{23}\\ {\text{A}}_{31}& {\text{A}}_{32}& {\text{A}}_{33}\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}-3& -1& -5\\ -1& 19& -21\\ 22& -12& -2\end{array}\right]}$.