Determinant is a scalar value that can be calculated from the elements of a square matrix, such as:

(\begin{array}{l}\left| \begin{matrix} a & b \ c & d \ \end{matrix} \right|.\end{array} )

For a 3×3 matrix, the determinant is determined by:

(\begin{array}{l}\begin{vmatrix} a_{1} & b_{1} & c_{1}\ a_{2}& b_{2} & c_{2}\ a_{3}& b_{3} & c_{3} \end{vmatrix}\end{array} )

= a1(b2c3 - b3c2) - b1(a2c3 - a3c2) + c1(a2b3 - a3b2).

In this article, we discussed the properties of determinants, multiplication of determinants and the determinants formula.

Table of Contents

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Determinants Introduction

Evaluation of the Determinant Using Sarrus Method

Symmetric and Skew Symmetric Determinants

Multiplication of Two Determinants

Introduction to Determinants

The development of determinants occurred when mathematicians were attempting to solve a system of simultaneous linear equations.

(\begin{array}{l}E.g.\left. \begin{matrix} {{a}_{1}}x+{{b}_{1}}y={{c}_{1}} \ {{a}_{2}}x+{{b}_{2}}y={{c}_{2}} \ \end{matrix} \right] \Rightarrow x=\frac{{{b}_{2}}{{c}_{1}}-{{b}_{1}}{{c}_{2}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}};;; \text{and} ;; y=\frac{{{a}_{1}}{{c}_{2}}-{{a}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\end{array} )

Mathematicians defined the symbol $\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} \ {{a}_{2}} & {{b}_{2}} \ \end{matrix} \right|$ as a determinant of order 2, with the four numbers arranged in rows and columns known as its elements. When written in this form, the horizontal lines are known as rows and the vertical lines as columns, with the shape of every determinant being a square. If a determinant is of order n, then it contains n rows and n columns.

E.g. $$\left| \begin{matrix} a_1 & b_1 \ a_2 & b_2 \ \end{matrix} \right|, \left| \begin{matrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \ \end{matrix} \right|$$ are determinants of second and third order respectively.