A factor which decisively affects the nature or outcome of something is called a Determinant. A quantity obtained by the addition of products of the elements of a square matrix according to a given rule.

Matrices: A, B, C Elements of a matrix: [latex]a_i, b_i, a_{ij}, b_{ij}, c_{ij}[/latex] Determinant of a matrix: det A Minor of an element [latex] a_{ij}: M_{ij}[/latex] Cofactor of an element [latex] a_{ij}: C_{ij}[/latex] Transpose of a matrix: [latex] A^T, \widetilde{A}[/latex] Adjoint of a matrix: adj A Trace of a matrix: tr A Inverse of a matrix: [latex]A^-1[/latex] Real number: k Real variables: [latex]x_i[/latex] Natural numbers: m, n
1. Second Order Determinant det A = [latex]\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{vmatrix}[/latex] = [latex]a_1 b_2 -a_2 b_1[/latex]
2. Third Order Determinant det A = [latex]\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{vmatrix}[/latex] = [latex]a_{11} a_{22} a_{33} + a_{12} a_{23} a_ {31} + a_{13} a_{21} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} - a_{13} a_ {22} a_{31}[/latex]
3. Third Order Determinant
4. N-th Order Determinant det A = [latex]\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2j} & \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \\ a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \\ a_{n1} & a_{n2} & \cdots & a_{nj} & \cdots & a_{nn}\\ \end{vmatrix}[/latex]
5. Minor The minor [latex]M_{ij}[/latex] associated with the element [latex]a_{ij}[/latex] of n-th order matrix A is the (n -1)-th order determinant derived from the matrix A by deletion of its i-th row and j-th column.
6. Cofactor [latex]C_{ij} = (-1)^{i+j} M_{ij}[/latex]
7. Laplace Expansion of n-th Order Determinant Laplace expansion by elements of the i-th row det A = [latex]\displaystyle\sum_{j=1}^{n} a_{ij} C_{ij}, i = 1, 2..., n[/latex] Laplace expansion by elements of the j-th column det A = [latex]\displaystyle\sum_{i=1}^{n}[/latex][latex]a_{ij} C_{ij}, i = 1, 2..., n[/latex]