# Formulas for Determinants of Matrices

## Determinant of a 2 by 2 Matrix

Let A be 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 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 3 by 3 matrix $A$ 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 forms for the determinant of $n \times n$ matrices for $n \gt 2$.