Introduction

Determinant is defined for square matrices. A square matrix has the same number of rows as columns, like a $2\times 2$ matrix $A$ or a $3\times 3$ matrix $B$.

$$A= \left[ \begin{array}{rr} a & b \\ c & d\\ \end{array}\ri... ...' &c''\\ a'' & b'' &c''\\ \end{array}\right] \hspace{2cm}$$

We will discuss few applications of determinant. The cross product of two vectors in $R^3$ can be defined using the determinant of a $3\times 3$ matrix. The area of the parallelogram generated by these two vectors can be be obtained using as a determinant. The volume of the parallelepiped formed by any three nonzero vectors in $R^3$, also can be find using determinant. Determinant is used in change of variable of integrals in calculus. It can be used in finding eigenvalues of matrix. Determinant of a matrix can tell us about invertible of the matrix, number of solutions of an a linear system of n-equations in n-unknowns, and many other applications.