Program to find Determinant of a Matrix

The determinant of a Matrix is defined as a special number that is defined only for square matrices (matrices that have the same number of rows and columns). A determinant is used in many places in calculus and other matrices related to algebra, it actually represents the matrix in terms of a real number which can be used in solving a system of a linear equation and finding the inverse of a matrix.

Determinant of 2 x 2 Matrix:

Determinant of 3 x 3 Matrix:

How to calculate?

The value of the determinant of a matrix can be calculated by the following procedure: 

• For each element of the first row or first column get the cofactor of those elements.
• Then multiply the element with the determinant of the corresponding cofactor. 
• Finally, add them with alternate signs. As a base case, the value of the determinant of a 1*1 matrix is the single value itself. 

The cofactor of an element is a matrix that we can get by removing the row and column of that element from that matrix.

30

Time Complexity: O(n3)
Space Complexity: O(n²), Auxiliary space used for storing cofactors.

Note: In the above recursive approach when the size of the matrix is large it consumes more stack size.

Determinant of a Matrix using Determinant properties

We calculates the determinant of an N x N matrix using Gaussian elimination and a series of transformations that reduce the matrix to upper triangular form.

• Converting the given matrix into an upper triangular matrix using determinant properties 
• The determinant of the upper triangular matrix is the product of all diagonal elements. 
• Iterating every diagonal element and making all the elements down the diagonal as zero using determinant properties 
• If the diagonal element is zero then search for the next non-zero element in the same column.

There exist two cases:

• Case 1: If there is no non-zero element. In this case, the determinant of a matrix is zero 
• Case 2: If there exists a non-zero element there exist two cases 
  • Case A: If the index is with a respective diagonal row element. Using the determinant properties make all the column elements down to it zero
  • Case B: Swap the row with respect to the diagonal element column and continue the Case A operation.

30

Time complexity: O(n³) 
Auxiliary Space: O(n), Space used for storing row.

Determinant of a Matrix 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 numpy.linalg.det(mat) which returns the determinant value of the matrix mat passed in the argument.