The method to determine the determinant of a square Matrix

Introduction

In Matrix Algebra a matrix is set of values arranged in rows and columns. Some matrices have equal number of columns and rows. However some matrices do not have this property. When the rows and columns are equal then that matrix is a square matrix.

Determinant is a number not a matrix and it is useful because this number indicates whether matrix can be invertible or not. If one wants to solve a system of linear equations using matrix method and if the number of equations is quite large it is sensible to calculate the determinant first to know whether it is zero or not. This with indicate whether the system of linear equations has an inverse matrix. If the determinant is zero then the matrix is singular. That is, the system of equation does not have any solution. If, the determinant is not zero then the matrix is non-singular. In other words, the system of equations can be inverted and there fore it has solution to the system of linear equations. That is, in Matrix Algebra it is useful to devise a method to calculate the determinant or devise an algorithm to calculate using a computer is advisable when one want to know for a considerable number of simultaneous linear equations to determine whether they have a deternininant value more than Zero or a negative real number.

Determinant of a 2*2 matrix

The 2*2 square matrix has two rows and two columns. The determinant of this matrix is the product of the diagonal element minus the product of the non diagonal elements. For example, one can determine the determinant of a 2*2 square matrix as follows:

Say the2*2 square matrix is | a b |

| c d |

This matrix determinant is = a*d – c*b. Where the main diagonal elements are “a and d” in the matrix and “c and d” are non-diagonal elements.

Method to determine the determinant of (n*n) square matrix

This method uses “minors” and “co-factors” in determining the determinant of any square matrix.

The definition of “minors” is as follows:

A minor for any element of a square matrix is the determinant that results from deleting the row and column of that element and the resulting square matrix.

The co-factor matrix is the alternate addition and subtraction of the minor matrix. The following example will shed light on these concepts.

Say one wants to know the Minor matrix of a square matrix. The square matrix is as follows:

| 3 2 4 |

| 5 8 2 |

| 4 7 6 |

In this matrix the number 3 is in the first row and first column. The corresponding minor matrix of element is | 8 2 |

| 7 6 |

Like wise for every element there exist a minor by deleting the row and column and finding the residual elements. For the above matrix the minors are as follows:

| 8 2 | | 5 2 | | 5 8 | | 2 4 | | 3 4 | | 3 2 | | 2 4 | | 3 4 | | 3 2 |

| 7 6 | | 4 6 | | 4 7 | | 7 6 | | 4 6 | | 4 7 | | 8 2 | | 5 2 | | 5 8 |

There fore the determinant of these minor matrices are as follows:

| 8 2 | the determinant is 8*6 – 7*2 = 48 – 14 = 34.

| 7 6 |

| 5 2 |

| 4 6 | the determinant is 5*6 – 4*2 = 30 – 8 = 22

| 5 8 |

| 4 7 | the determinant is5*7 – 4*8 = 35 – 32 = 3

| 2 4 |

| 7 6 | the determinant is 2*6 – 4*7 = 12 – 28 = -16

| 3 4 |

| 4 6 | the determinant is 3*6 – 4*4 = 18 – 16 = 2

| 3 2 |

| 4 7 | the determinant is 3*7 – 4*2= 21 – 8 = 13

| 2 4 |

| 8 2 | the determinant is 2*2 – 8*4 = 4 – 32 = -28

| 3 4 |

| 5 2 | the determinant is 3*2 – 5*4 = 6 – 20 = -14

| 3 2 |

| 5 8 | the determinant is 3*8 – 5*2 = 24 – 10 = 14.

There fore, the minor matrix according to the definition of minors as determinant of minor matrices of the above matrix is as follows:

| 34 22 3 |

|-16 2 13 |

|-28 -14 14 |

There fore the co-factor matrix is the addition and subtraction of the minor matrix. In this instant the co-factor matrix is as follows:

| 34 -22 3 |

| 16 2 -13 |

|-28 14 14 |

The determinant of the above matrix is the multiplication of any row or column by its co-factor and adding it. In this instant say if I take the 1st row of the above matrix, that is | 3 2 4 | then multiply by the co-factor of this row. That is, multiply by | 34 -22 3 |, which is the 1st row of the co-factor matrix.

There fore the determinant of the above matrix = 3*34 -22*2 + 4*3 = 102 – 44 + 8 = 64.

Determinant of (n*n) square matrix

Pick the column or row with more zeros.

Calculate the minors and then the co-factors of the non-zero elements

Multiply the co-factor with the elements to find the determinant.

The relationship between co-factors, determinant, adjoint and inverse of a matrix

Adjoint is the transpose matrix of co-factor matrix. If a matrix has a determinant then the inverse of a matrix can be found by dividing the elements of adjoint matrix by the determinant. In this sense, the minor matrix and co-factors are useful in the sense it can be used to find the inverse of a matrix particularly if it has a determinant.

In the above example the co-factor matrix is | 34 -22 3 |

| 16 2 -13|

|-28 14 14 |

There fore the transpose matrix of the co-factor or adjoint matrix is as follws:

| 34 16 -28|

|-22 2 14|

| 3 -13 14|

There fore the inverse is the adjoint elements divided by the determinant. There fore, the inverse of the above mentioned matrix is as follows:

| 34/60 16/60 -28/60 |

|-22/60 2/60 14/60|

| 3/60 -13/60 14/60|

It is also important to note that the determinant of a diagonal matrix or a triangular matrix is the product of the main diagonal elements. There fore if any matrix can be transformed in to triangualr or diagonal matrix by elementary row operations may be a useful method in conjunction with the above concepts of minors and co-factors in determining the determinants of a matrix.

As discussed above, the concepts of minors, co-factors and adjoints are useful concepts to be applied to any square matrix to determine the determinants and inverse of a matrix if there exist a determinant.