Definition: Every square matrix can be associated to an expression or a number which is known as its determinant. If A = [aij] is a square matrix of order n then the determinant of A is denoted by or |A| or \(\left| \begin{matrix}{{a}_{11}} & {{a}_{12}} & {{a}_{13}} & . & {{a}_{ij}} & . & {{a}_{1n}} \\{{a}_{21}} & {{a}_{22}} & {{a}_{23}} & . & {{a}_{2j}} & . & {{a}_{2n}} \\. & . & . & . & . & . & . \\{{a}_{i1}} & {{a}_{i2}} & {{a}_{i3}} & . & {{a}_{ij}} & . & {{a}_{in}} \\. & . & . & . & . & . & . \\{{a}_{n1}} & {{a}_{n2}} & {{a}_{n3}} & . & {{a}_{nj}} & . & {{a}_{nm}} \\\end{matrix} \right|\).
Determinant of a square matrix of order 1: If A = [a11] is a square matrix of order 1, then the determinant of A is defined as |A| = a11 or |a11|= a11.
Determinant of a square matrix of order 2: If \(A=\left[\begin{matrix}{{a}_{11}}&{{a}_{12}} \\{{a}_{21}}&{{a}_{22}} \\\end{matrix}\right]\) is a matrix of order 2, then the expression a11a22 – a12 a21 is defined as the determinant of A i.e., \(\left| A\right|=\left|\begin{matrix}{{a}_{11}}&{{a}_{12}}\\{{a}_{21}}&{{a}_{22}}\\\end{matrix}\right|\)= a₁₁ a₂₂ – a₁₂ a₂₁.
The determinant of a square matrix of order 2 is equal to the product of the diagonal elements minus the product of the diagonal elements.
Determinant of square matrix of order 3: If \(A=\left[ \begin{matrix}{{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\{{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\{{a}_{31}} & {{a}_{23}} & {{a}_{33}} \\\end{matrix} \right]\) is a square matrix of order 3, then the expression a₁₁a₂₂a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₃₂ a₂₁ – a₁₁ a₂₃ a₃₂ – a₂₂ a₁₃ a₃₁ – a₁₂ a₂₁ a₃₃ is defined as the determinant of A.
\(\left| A \right|=\left| \begin{matrix}{{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\{{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\{{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\end{matrix}\right|={{a}_{11}}\left| \begin{matrix}{{a}_{22}} & {{a}_{23}} \\{{a}_{32}} & {{a}_{33}} \\\end{matrix} \right|-{{a}_{12}}\left| \begin{matrix}{{a}_{21}} & {{a}_{23}} \\{{a}_{31}} & {{a}_{33}} \\\end{matrix} \right|+{{a}_{13}}\left| \begin{matrix}{{a}_{21}} & {{a}_{22}} \\{{a}_{31}} & {{a}_{32}} \\\end{matrix} \right|\).
Example: Evaluate determinant of \(A=\left[ \begin{matrix}3 & -2 & 4 \\1 & 2 & 1 \\0 & 1 & -1 \\\end{matrix} \right]\).
Solution: \(\left| A \right|=\left| \begin{matrix}3 & -2 & 4 \\1 & 2 & 1 \\0 & 1 & -1 \\\end{matrix} \right|\).
= \(3\left| \begin{matrix}2 & 1 \\1 & -1 \\\end{matrix} \right|-\left( -2 \right)\left| \begin{matrix}1 & 1 \\0 & -1 \\\end{matrix} \right|+4\left| \begin{matrix}1 & 2 \\ 0 & 1 \\\end{matrix} \right|\).
= 3 (-2 -1) + 2 (-1 -0) + 4 (1 – 0)
= -9 – 2 + 4
= – 7
Only square matrices have determinants. The matrices which are not square do not have determinant.
The determinant of a square matrix of order 3 can be expanded along any row or column.
If a row or a column of a determinant consists of all zeros, then the value of the determinant is zero.
The determinant of a skew symmetric matrix of odd order is zero.
Ex: \(\left| \begin{matrix}0 & b & -c \\-b & 0 & a \\c & -a & 0 \\\end{matrix} \right|=0\).
A determinant is called circulant if its rows (columns) are cyclic shifts of the first row (columns).
Ex: \(\left| \begin{matrix}a & b & c \\b & c & a \\c & a & b \\\end{matrix} \right|\). It can be show that its value is 3abc – a₃ – b₃ – c₃.