What is Determinant of a Matrix?
Determinant of a Matrix is a special number that is defined only for square matrices (matrices which have same number of rows and columns). Determinant is used at many places in calculus and other matrix related algebra, it actually represents the matrix in term of a real number which can be used in solving system of linear equation and finding the inverse of a matrix.
How to calculate?
The value of determinant of a matrix can be calculated by following procedure –
For each element of first row or first column get cofactor of those elements and then multiply the element with the determinant of the corresponding cofactor, and finally add them with alternate signs. As a base case the value of determinant of a 1*1 matrix is the single value itself.
Cofactor of an element, is a matrix which we can get by removing row and column of that element from that matrix.
Determinant of 2 x 2 Matrix:
Determinant of 3 x 3 Matrix:
Determinant of the matrix is : 30
Adjoint and Inverse of a Matrix
There are various properties of the Determinant which can be helpful for solving problems related with matrices,
This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
In Above Method Recursive Approach is discussed.When the size of matrix is large it consumes more stack size
In this Method We are using the properities of Determinant.In this approach we are converting the given matrix into upper triangular matrix using determinant properties The determinant of upper traingular matrix is the product of all diagonal elements For properties on determinant go through this website https://cran.r-project.org/web/packages/matlib/vignettes/det-ex1.html
In this approach we are iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties
If the diagonal element is zero then we will search next non zero element in the same column
There exist two cases
If there is no non zero element.In this case the determinant of matrix is zero
If there exists non zero element there exist two cases
if index is with respective diagonal row element.Using the determinant properties we make all the column elements down to it as zero
Here we need to swap the row with respective to diagonal element column and continue the case ‘a; operation
Below is the implementation of the above approach:
Determinant of the matrix is : 30
Time complexity : O(n3)
Auxiliary Space : O(n)
Method 3 : Using numpy package in python
There is a built in function or method in linalg module of numpy package in python. It can be called as numpy.linalg.det(mat) which returns the determinant value of matrix mat passed in the arguement.
Determinant of the matrix is: 30.0
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