MATRIX Course

Cofactor matrix

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The cofactor matrix of $A$, called co-matrice/Com(A) in French, is the matrix of the cofactors.

\[ C = \begin{bmatrix} C_{11}&C_{12}&\cdots &C_{1n}\\ C_{21}&C_{22}&\cdots &C_{2n}\\ \vdots &\vdots &\ddots &\vdots \\ C_{n1}&C_{n2}&\cdots &C_{nn} \end{bmatrix} \]

Once you created this matrix, the inverted matrix formula is

\[ A^{-1} = \frac{1}{det(A)} C^T \]

Example

Given the following invertible matrix $A$ since $det(A)=66$. Using the cofactor matrix, solve $A^{-1}$.

\[ A=\begin{pmatrix} 3 & -2 & 4 \\ 2 & -4 & 5 \\1 & 8 & 2\end{pmatrix} \]
\[ C = \begin{pmatrix} + det(\begin{pmatrix}-4 & 5 \\ 8 & 2\end{pmatrix}) & - det(\begin{pmatrix}2 & 5 \\ 1 & 2\end{pmatrix}) & + det(\begin{pmatrix}2 & -4 \\ 1 & 8\end{pmatrix}) \\ - det(\begin{pmatrix}-2 & 4 \\ 8 & 2\end{pmatrix}) & + det(\begin{pmatrix}3 & 4 \\ 1 & 2\end{pmatrix}) & - det(\begin{pmatrix}3 & -2 \\ 1 & 8\end{pmatrix}) \\ + det(\begin{pmatrix}-2 & 4 \\ -4 & 5\end{pmatrix}) & - det(\begin{pmatrix}3 & 4 \\ 2 & 5\end{pmatrix}) & + det(\begin{pmatrix}3 & -2 \\ 2 & -4\end{pmatrix}) \end{pmatrix} \] \[ \Leftrightarrow C = \begin{pmatrix} -48 & 1 & 20 \\ 36 & 2 & -26 \\ 6 & -7 & -8 \\ \end{pmatrix} \] And \[ C^T = \begin{pmatrix} -48 & 36 & 6 \\ 1 & 2 & -7 \\ 20 & -26 & -8 \\ \end{pmatrix} \]

Then we have

\[ \begin{split} A^{-1} = \frac{1}{-66} * \begin{pmatrix} -48 & 36 & 6 \\ 1 & 2 & -7 \\ 20 & -26 & -8 \\ \end{pmatrix} \\ \Leftrightarrow \begin{pmatrix} 48/66 & -36/66 & -6/66 \\ -1/66 & -2/66 & 7/66 \\ -20/66 & 26/66 & 8/66 \\ \end{pmatrix} \Leftrightarrow \begin{pmatrix} 8/11 & -6/11 & -1/11 \\ -1/66 & -1/33 & 7/66 \\ -10/33 & 13/33 & 4/33 \\ \end{pmatrix} \end{split} \]

Code in R

library('MASS') # fractions
library('matlib') # rowCofactors

# I don't know how we can find Ct in R
# but you got some code here
# https://stackoverflow.com/questions/29046934/calculate-matrix-of-cofactors-in-r
# here is my version
# takes a matrix, return the cofactor matrix
cofactorMatrix <- function (A){
    n <- dim(A)[1]
    values <- NULL
    for (i in 1:n) {
        values <- c(values, rowCofactors(A, i))
    }
    return(matrix(values, n, n, byrow = TRUE))
}

A <- matrix(c(3,2,1,-2,-4,8,4,5,2), nrow = 3, ncol = 3)
C <- matrix(cofactorMatrix(A), nrow = 3, ncol = 3)
A.inv <- t(C) * (-1/66)

# print (as fractions)
fractions(A.inv)
#       [,1]   [,2]   [,3]
# [1,]   8/11  -6/11  -1/11
# [2,]  -1/66  -1/33   7/66
# [3,] -10/33  13/33  4/33