What is Determinant of a matrix? How to calculate determinant of Order 1, 2, 3 and arbitrary Order? What are the Properties of Determinants?

Determinants

Introduction

        Each n-square matrix A =[aij]is assigned a special scalar called the determinant of A, denoted by det (A) or |A|or

            

          An n✕n array of scalars enclosed by straight lines, called a determinant of order n, is not a matrix but denotes the determinant of the enclosed array of scalars, i.e., the enclosed matrix.

         The determinant function was first discovered during the investigation of systems of linear equations. The determinant is an indispensable tool in investigating and obtaining properties of square matrices.

 

Determinants of Order 1 and 2

The Determinants of Order 1 and 2 are defined as follows:

The determinant of a 1x1 matrix A =[a11] is the scalar a11 itself, that is, 

                                        

The determinant of order two may easily remembered by using the following diagram:

                                                            

That is, the determinant is equal to the product of the elements along the one plus-labeled arrow minus the product of the elements along the one minus-labeled arrow.

Example 1: Find the determinant of matrix A.

Solution:



Example 2: Find the determinant of matrix B.

Solution:

Determinants of Order 3 

    There are many ways on finding the determinant of a 3x3 matrix. 

To calculate the determinant of a 3x3 matrix, click the following:

Sarrus Rule
Triangle's Rule
Co-Factor Expansion along the Column
Co-Factor Expansion along the Row
Row Echelon Form

Properties of Determinants

Theorem: The determinants of a matrix A and its transpose Aᵀ are equal, that is,  

|A| = |Aᵀ|

Theorem: Let A be a square matrix.

i. If A has a row (column) of zeros, then |A| = 0. 

ii. If A has two identical rows (columns), then |A| = 0. 

iii. If A is triangular, i.e. A has zeros above or below the diagonal, then A = product of diagonal elements. Thus, in particular |I|=1 where I is identity matrix. 

Theorem: Suppose B is obtained from A by an elementary row (column) operation.

i. If two rows (columns) of A were interchanged, then |B|= −|A|. 

ii. If a row (column) of A were multiplied by a scalar k, then |B|= k|A|.

iii. If a multiple of a row (column) of A were added to another row (column) of A, then |B|= |A|. 

Theorem: The determinant of a product of two matrices A and B is the product of their determinants, that is,

det(AB) = det(A) det (B)

Note: Base on the theorem above, the determinant is a multiplicative function.

Theorem: Let A be a n-square matrix. Then the following are equivalent:

i. A is invertible, that is, A has an inverse A⁻¹.
ii. AX = 0 has only the zero solution.
iii. The determinant of A is not zero, that is det(A) ≠ 0..