Determinant and Cofactors

16.1 Formula for \(|A|\)

For a \(2 \times 2\) matrix \(A\), the formula for \(|A|\) can be derived as follows:

\[\begin{align} |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix}= \begin{vmatrix} a & 0 \\ c & d \end{vmatrix}+\begin{vmatrix} 0 & b \\ c & d \end{vmatrix} \end{align}\]\[\begin{align} =\begin{vmatrix} a & 0 \\ c & 0 \end{vmatrix}+\begin{vmatrix} a & 0 \\ 0 & d \end{vmatrix}+\begin{vmatrix} 0 & b \\ c & 0 \end{vmatrix}+\begin{vmatrix} 0 & b \\ 0 & d \end{vmatrix} \end{align}\]\[\begin{align} =0+ad-bc+0=ad-bc \end{align}\]

For a \(3 \times 3\) matrix, first row can be seperated into \(3\) pieces as demonstrated above. For each of the individual separated matrices, the second row will be separated into \(3\) pieces giving a total of \(9\) matrices. Finally, for each of these \(9\) matrices, the third row will be separated into \(3\) pieces, giving a total of \(27\) matrices. Out of these \(27\) matrices, a lot will have \(0\) determinant. Matrices with non-zero determinant will have one entry from each row and column. The splitted matrices with non-zero determinant for a \(3 \times 3\) matrix \(A\) is shown below. The sign of individual determinants is derived based on number of row exchanges needed to get a diagonal matrix.

\[\begin{align} |A| = \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} \end{align}\]\[\begin{align} =\begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33} \end{vmatrix}+\begin{vmatrix} a_{11} & 0 & 0\\ 0 & 0 & a_{23}\\ 0 & a_{32} & 0 \end{vmatrix}+\begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & 0\\ 0 & 0 & a_{33} \end{vmatrix} \end{align}\]\[\begin{align} +\begin{vmatrix} 0 & a_{12} & 0\\ 0 & 0 & a_{23}\\ a_{31} & 0 & 0 \end{vmatrix}+\begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & 0 & 0\\ 0 & a_{32} & 0 \end{vmatrix}+\begin{vmatrix} 0 & 0 & a_{13}\\ 0 & a_{22} & 0\\ a_{31} & 0 & 0 \end{vmatrix} \end{align}\]\[\begin{align} =a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}+ \\ a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} \end{align}\]

The generic formula for a \(n \times n\) matrix \(A\) is as follows:

\[\begin{align} |A| = \sum_{\text{n! terms}} \pm a_{1\alpha}a_{2\beta}...a_{n\omega} \end{align}\]\[\begin{align} \text{such that}: (\alpha, \beta, ..., \omega) = \text{Permutation of}(1,2,3,...,n) \end{align}\]

16.2 Cofactors

Cofactor Formula connects determinant of \(n \times n\) matrix to the determinant of smaller matrix of size \(n-1 \times n-1\). The determinant of the \(3 \times 3\) matrix \(A\) shown in cofactor format is as follows:

\[\begin{align} |A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) + a_{12}(-a_{21}a_{33} + a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \end{align}\]

The above determinant formula can be seen as the combination of following permutations.

\[\begin{align} \begin{vmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & a_{23}\\ 0 & a_{32} & a_{33} \end{vmatrix};\begin{vmatrix} 0 & a_{12} & 0\\ a_{21} & 0 & a_{23}\\ a_{31} & 0 & a_{33} \end{vmatrix};\begin{vmatrix} 0 & 0 & a_{13}\\ a_{21} & a_{22} & 0\\ a_{31} & a_{32} & 0 \end{vmatrix} \end{align}\]\[\begin{align} \text{Cofactor of } a_{ij} = C_{ij} = (-1)^{i+j}|n-1 \text{ matrix with row } i \text{ col } j \text{ erased}| \end{align}\]

Cofactors without the sign are called Minors. Hence,

\[\begin{align} |A| = a_{11}C_{11} + a_{12}C_{12} + ... + a_{1n}C_{1n} \end{align}\]