# Determinants Formulas

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# Determinants Formulas

### Introduction

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.

### Formulae

Matrices: A, B, C
Elements of a matrix: $$a_i, b_i, a_{ij}, b_{ij}, c_{ij}$$
Determinant of a matrix: det A
Minor of an element $$a_{ij}: M_{ij}$$
Cofactor of an element $$a_{ij}: C_{ij}$$
Transpose of a matrix: $$A^T, \widetilde{A}$$
Trace of a matrix: tr A
Inverse of a matrix: $$A^-1$$
Real number: k
Real variables: $$x_i$$
Natural numbers: m, n

1. Second Order Determinant
det A = $$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \\ \end{vmatrix}$$ = $$a_1 b_2 -a_2 b_1$$

2. Third Order Determinant
det A = $$\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\\ \end{vmatrix}$$ = $$a_{11} a_{22} a_{33} + a_{12} a_{23} a_ {31} + a_{13} a_{21} a_{31} – a_{11} a_{23} a_{32} – a_{12} a_{21} a_{33} – a_{13} a_ {22} a_{31}$$

3. Third Order Determinant

4. N-th Order Determinant
det A = $$\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2j} & \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \\ a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \\ a_{n1} & a_{n2} & \cdots & a_{nj} & \cdots & a_{nn}\\ \end{vmatrix}$$

5. Minor
The minor $$M_{ij}$$ associated with the element $$a_{ij}$$ 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
$$C_{ij} = (-1)^{i+j} M_{ij}$$

7. Laplace Expansion of n-th Order Determinant
Laplace expansion by elements of the i-th row
det A = $$\displaystyle\sum_{j=1}^{n} a_{ij} C_{ij}, i = 1, 2…, n$$
Laplace expansion by elements of the j-th column
det A = $$\displaystyle\sum_{i=1}^{n}$$$$a_{ij} C_{ij}, i = 1, 2…, n$$