Cofactor Matrix

A pivotal concept in matrix operations, the cofactor matrix C is a transformation of a given square matrix M. This transformation process involves replacing each individual element M[i][j] of the matrix M with its respective cofactor C[i][j].

As a reminder, the cofactor of an element M[i][j] is determined by this specific formula:

$$ C[i][i] = (-1)^{(i+j)} \cdot \det(M[i][j]) $$

Here, M[i][j] refers to the submatrix derived from the original matrix A upon removing the ith row and the jth column.

In essence, the cofactor is the determinant of the resultant submatrix M[i][j] when the corresponding row and column of the element are excluded.

Let's elucidate this with a concrete example.

Suppose we have the following matrix:

$$ M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$

The element located at the first row (i=1) and first column (j=1) holds the value "1".

To find its cofactor C[1][1], we first form the submatrix M[1][1] by excluding the row and column where the element resides.

 

We then compute the determinant of this submatrix M[1][1], and multiply the result by (-1)^i+j

$$ C[1][1]= (-1)^{(i+j)} \cdot \det(M[i][j]) $$

Upon substituting i=1 and j=1, we find:

$$ C[1][1]= (-1)^{(i+j)} \cdot \det(M[1][1]) $$

$$ C[1][1]= (-1)^{(1+1)} \cdot \det \begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} $$

$$ C[1][1]= (-1)^2 \cdot ( 5 \cdot 9 - 6 \cdot 8 ) $$

$$ C[1][1]= 1 \cdot ( 45 - 48 ) $$

$$ C[1][1]= 1 \cdot ( - 3 ) $$

$$ C[1][1]= -3 $$

This process yields the cofactor C[1][1] of the element M[1][1].

With this, we start to populate the cofactor matrix of M:

$$ C_M = \begin{pmatrix} C[1][1] & C[1][2] & C[1][3] \\ C[2][1] & C[2][2] & C[2][3] \\ C[3][1] & C[3][2] & C[3][3] \end{pmatrix} $$

In this matrix, each C[i][j] is the cofactor of its corresponding element M[i][j].

By substituting C[1][1] = -3, we obtain:

$$ C_M = \begin{pmatrix} -3 & C[1][2] & C[1][3] \\ C[2][1] & C[2][2] & C[2][3] \\ C[3][1] & C[3][2] & C[3][3] \end{pmatrix} $$

Subsequently, we repeat the calculation of the cofactor C[i][j] for all remaining elements of the matrix M, progressively filling out the cofactor matrix C_M.

$$ C_M = \begin{pmatrix} \det \begin{pmatrix} 5 & 6 \\ 8 & 9\end{pmatrix} & - \det \begin{pmatrix} 4 & 6 \\ 7 & 9\end{pmatrix} & \det \begin{pmatrix} 4 & 5 \\ 7 & 8\end{pmatrix} \\ - \det \begin{pmatrix} 2 & 3 \\ 8 & 9\end{pmatrix} & \det \begin{pmatrix} 1 & 3 \\ 7 & 9\end{pmatrix} & - \det \begin{pmatrix} 1 & 2 \\ 7 & 8\end{pmatrix} \\ \det \begin{pmatrix} 2 & 3 \\ 5 & 6\end{pmatrix} & - \det \begin{pmatrix} 1 & 3 \\ 4 & 6\end{pmatrix} & \det \begin{pmatrix} 1 & 2 \\ 4 & 5 \end{pmatrix} \end{pmatrix} $$

$$ C_M = \begin{pmatrix} 45-48 & - (36 - 42) & 32 - 35 \\ - ( 18-24) & 9 - 21 & - ( 8 - 14) \\ 12 - 15 & - (6-12) & 5 - 8 \end{pmatrix} $$

Upon completion, we have the final cofactor matrix of M:

$$ C_M = \begin{pmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{pmatrix} $$

Now, one might ask about the relevance of the cofactor matrix. It's essential to note that the cofactor matrix is an integral element in calculating a matrix's inverse. In fact, if an inverse for a matrix M exists, it can be evaluated as the transpose of the cofactor matrix C of M, divided by the determinant of M: $$ M^{-1} = \frac{1}{\det(M)} \cdot C_M^T $$

This underscores the foundational importance of the cofactor matrix in linear algebra.