Formulas for Matrix Determinants

Determinant of a 2 by 2 Matrix

Let \( A \) be a 2 by 2 matrix:
\( A = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \end{bmatrix} \)
The determinant of matrix \( A \) is denoted by \( |A| \) and is given by:
\( |A| = \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \end{vmatrix} = a_1 b_2 - a_2 b_1 \)

Determinant of a 3 by 3 Matrix

Let \( A \) be a 3 by 3 matrix given by:
\( A = \begin{bmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \)
The determinant of matrix \( A \) is given by:
\( |A| = \begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \end{vmatrix} \)
There are many other methods and formulas for the determinants of \( n \times n \) matrices for \( n \gt 2 \).

Formulas for Matrix Determinants

Determinant of a 2 by 2 Matrix

Determinant of a 3 by 3 Matrix

More References and Links