Computing Determinants

We have explicit explicit expressions to compute the determinants in terms of the elements of a matrix only for \(1 \times{1}\) and \(2 \times{2}\) matrices. For n = 1,

\[\det(A) = \det([a_{11}]) = a_{11}\]

For n = 2,

\[\det(A) = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} = ad - bc \]

For matrices of higher orders \(A \in \mathbb{R}^{n \times{n}}\), we recursively try to reduce the problem of finding the determinant of the \(n \times{n}\) matrix to finding the determinant of (\(n-1) \times(n-1)\) matrices until the problem gets reduced to computing the determinants of (many) simple \(2 \times{2}\) matrices. This algorithm of finding the determinant of any general square matrix is called the Laplace Expansion. For a matrix \(A \in \mathbb{R}^{n \times{n}}\), Laplace expansion can either be done along rows or along columns:

1. Expansion along row i

\[\det(A) = \sum_{j=1}^{n} (-1)^{(i + j)} a_{ij} \times \det(A_{i,j})\]

2. Expansion along column j

\[\det(A) = \sum_{i=1}^{n} (-1)^{(i + j)} a_{ij} \times \det(A_{i,j})\]

Here \(A_{i,j}\) is the submatrix of \(A\) that is obtained on deleting the \(i^{th}\) row and the \(j^{th}\) column. The determinant of this matrix \(\det(A_{i,j})\) is called the minor of the term \(a_{ij}\) and is usually denoted by \(M_{i,j}\). And when the minor is multiplied by the term \((-1)^{(i+j)}\) it's called the cofactor of \(a_{ij}\) denoted by \(C_{i,j}\).

Example

Let us try to compute the determinant of

\[A = \begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 0 & 0 & 1 \\ \end{vmatrix} \]

Using the Laplace expansion along the first row we get

\[\det(A) = (-1)^{(1+1)} . 1\begin{vmatrix} 1 & 2 \\ 0 & 1 \\ \end{vmatrix} + (-1)^{(1+2)} . 2\begin{vmatrix} 3 & 2 \\ 0 & 1 \\ \end{vmatrix} + (-1)^{(1+3)} . 3\begin{vmatrix} 3 & 1 \\ 0 & 0 \\ \end{vmatrix} \]

Now we can apply the determinant formula for \(2\times2\) matrices and obtain the final result

\[\det(A) = 1(1-0) - 2(3-0) + 3(0-0) = -5\]