Find the cofactor matrix of

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

Transpose and Adjoint of a Matrix

Question

Find the cofactor matrix of $[\begin{array}{ccc} 2 & - 3 & 3 \\ 2 & 2 & 3 \\ 3 & - 2 & 2 \end{array}]$

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[\begin{array}{ccc} 2 & - 3 & 3 \\ 2 & 2 & 3 \\ 3 & - 2 & 2 \end{array}]

M_{11} = ∣ \begin{array}{cc} 2 & 3 \\ - 2 & 2 \end{array} ∣ = 4 + 6 = 10 \Rightarrow C_{11} = {(- 1)}^{1 + 1} M_{11} = 1 \times 10 = 10

M_{12} = ∣ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} ∣ = 4 - 9 = - 5 \Rightarrow C_{12} = {(- 1)}^{1 + 2} M_{12} = - 1 \times - 5 = 5

M_{13} = ∣ \begin{array}{cc} 2 & 2 \\ 3 & - 2 \end{array} ∣ = - 4 - 6 = - 10 \Rightarrow C_{13} = {(- 1)}^{1 + 3} M_{13} = 1 \times - 10 = - 10

M_{21} = ∣ \begin{array}{cc} - 3 & 3 \\ - 2 & 2 \end{array} ∣ = - 6 + 6 = 0 \Rightarrow C_{21} = {(- 1)}^{1 + 1} M_{21} = 1 \times 0 = 0

M_{22} = ∣ \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} ∣ = 4 - 9 = - 5 \Rightarrow C_{22} = {(- 1)}^{2 + 2} M_{22} = 1 \times - 5 = - 5

M_{23} = ∣ \begin{array}{cc} 2 & - 3 \\ 3 & - 2 \end{array} ∣ = - 4 - 9 = - 13 \Rightarrow C_{23} = {(- 1)}^{2 + 3} M_{23} = - 1 \times - 13 = 13

M_{31} = ∣ \begin{array}{cc} - 3 & 3 \\ 2 & 3 \end{array} ∣ = - 9 - 6 = - 15 \Rightarrow C_{31} = {(- 1)}^{3 + 1} M_{31} = 1 \times - 15 = - 15

M_{32} = ∣ \begin{array}{cc} 2 & 3 \\ 2 & 3 \end{array} ∣ = 6 - 6 = 0 \Rightarrow C_{32} = {(- 1)}^{3 + 2} M_{32} = - 1 \times 0 = 0

M_{33} = ∣ \begin{array}{cc} 2 & - 3 \\ 2 & 2 \end{array} ∣ = 4 + 6 = 10 \Rightarrow C_{33} = {(- 1)}^{3 + 3} M_{33} = 1 \times 10 = 10

Co-factor matrix

C = [\begin{array}{ccc} 10 & 5 & - 10 \\ 0 & - 5 & 13 \\ - 15 & 0 & 10 \end{array}]