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 \( n \gt 2 \).

Formulas for Determinants of Matrices

Determinant of a 2 by 2 Matrix

Determinant of a 3 by 3 Matrix

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