The determinant is a special scalar value that can be computed from a square matrix. It is denoted as \(\det(A)\) or \(|A|\).

For a 2x2 Matrix:

The formula is simple: \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \)

For a 3x3 Matrix (and larger):

The determinant is found using cofactor expansion. We expand along a row or column (this calculator uses the first row). For each element in the row, we multiply it by the determinant of the smaller matrix that remains after removing the element's row and column. The signs of the terms alternate.

\( \det(A) = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - b\begin{vmatrix} d & f \\ g & i \end{vmatrix} + c\begin{vmatrix} d & e \\ g & h \end{vmatrix} \)