﻿ Determinant of a 5x5 Matrix Calculator

# Example of Finding the Determinant of a 5x5 Matrix

### This solution has been made using the calculator presented on the site.

Let's calculate the determinant A using a elementary transformations.
 det A = -2 7 0 6 -2 = 1 -1 3 2 2 3 4 0 5 3 2 5 -4 -2 2 0 3 -1 1 -4
The elements of column 1 multiplied by -1 are added to the corresponding elements of column 5.   more info
 -2 7 0 6 -2 + ( -2) * ( -1) 1 -1 3 2 2 + 1 * ( -1) 3 4 0 5 3 + 3 * ( -1) 2 5 -4 -2 2 + 2 * ( -1) 0 3 -1 1 -4 + 0 * ( -1)
This elementary transformation does not change the value of the determinant.
 = -2 7 0 6 0 = 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 0 3 -1 1 -4
The elements of row 2 multiplied by 4 are added to the corresponding elements of row 5.   more info
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 0 + 1 * 4 3 + ( -1) * 4 -1 + 3 * 4 1 + 2 * 4 -4 + 1 * 4
This elementary transformation does not change the value of the determinant.
 = -2 7 0 6 0 = 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
Row number 1
Column number 5
Element Row 1 and column 5
have been deleted
( -1) 1 + 5 * 0 *
 1 -1 3 2 3 4 0 5 2 5 -4 -2 4 -1 11 9
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
Row number 2
Column number 5
Element Row 2 and column 5
have been deleted
( -1) 2 + 5 * 1 *
 -2 7 0 6 3 4 0 5 2 5 -4 -2 4 -1 11 9
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
Row number 3
Column number 5
Element Row 3 and column 5
have been deleted
( -1) 3 + 5 * 0 *
 -2 7 0 6 1 -1 3 2 2 5 -4 -2 4 -1 11 9
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
Row number 4
Column number 5
Element Row 4 and column 5
have been deleted
( -1) 4 + 5 * 0 *
 -2 7 0 6 1 -1 3 2 3 4 0 5 4 -1 11 9
 -2 7 0 6 0 1 -1 3 2 1 3 4 0 5 0 2 5 -4 -2 0 4 -1 11 9 0
Row number 5
Column number 5
Element Row 5 and column 5
have been deleted
( -1) 5 + 5 * 0 *
 -2 7 0 6 1 -1 3 2 3 4 0 5 2 5 -4 -2
Products are summed. If the element is zero than product is zero too.
 = ( -1) 2 + 5 * 1 * -2 7 0 6 = 3 4 0 5 2 5 -4 -2 4 -1 11 9
 = - -2 7 0 6 = 3 4 0 5 2 5 -4 -2 4 -1 11 9
 -2 7 0 6 3 4 0 5 2 + ( -2) 5 + 7 -4 + 0 -2 + 6 4 -1 11 9
This elementary transformation does not change the value of the determinant.
 = - -2 7 0 6 = 3 4 0 5 0 12 -4 4 4 -1 11 9
The elements of column 3 multiplied by 3 are added to the corresponding elements of column 2.   more info
 -2 7 + 0 * 3 0 6 3 4 + 0 * 3 0 5 0 12 + ( -4) * 3 -4 4 4 -1 + 11 * 3 11 9
This elementary transformation does not change the value of the determinant.
 = - -2 7 0 6 = 3 4 0 5 0 0 -4 4 4 32 11 9
 -2 7 0 6 + 0 3 4 0 5 + 0 0 0 -4 4 + ( -4) 4 32 11 9 + 11
This elementary transformation does not change the value of the determinant.
 = - -2 7 0 6 = 3 4 0 5 0 0 -4 0 4 32 11 20
 -2 7 0 6 3 4 0 5 0 0 -4 0 4 32 11 20
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
 7 0 6 4 0 5 32 11 20
 -2 7 0 6 3 4 0 5 0 0 -4 0 4 32 11 20
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
 -2 0 6 3 0 5 4 11 20
 -2 7 0 6 3 4 0 5 0 0 -4 0 4 32 11 20
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * -4 *
 -2 7 6 3 4 5 4 32 20
 -2 7 0 6 3 4 0 5 0 0 -4 0 4 32 11 20
Row number 3
Column number 4
Element Row 3 and column 4
have been deleted
( -1) 3 + 4 * 0 *
 -2 7 0 3 4 0 4 32 11
Products are summed. If the element is zero than product is zero too.
 = - ( ( -1) 3 + 3 * ( -4) * -2 7 6 ) = 3 4 5 4 32 20
 = 4 * -2 7 6 = 3 4 5 4 32 20
The elements of row 2 multiplied by -4 are added to the corresponding elements of row 3.   more info
 -2 7 6 3 4 5 4 + 3 * ( -4) 32 + 4 * ( -4) 20 + 5 * ( -4)
This elementary transformation does not change the value of the determinant.
 = 4 * -2 7 6 = 3 4 5 -8 16 0
The elements of column 1 multiplied by 2 are added to the corresponding elements of column 2.   more info
 -2 7 + ( -2) * 2 6 3 4 + 3 * 2 5 -8 16 + ( -8) * 2 0
This elementary transformation does not change the value of the determinant.
 = 4 * -2 3 6 = 3 10 5 -8 0 0
 -2 3 6 3 10 5 -8 0 0
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * -8 *
 3 6 10 5
 -2 3 6 3 10 5 -8 0 0
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
 -2 6 3 5
 -2 3 6 3 10 5 -8 0 0
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
 -2 3 3 10
Products are summed. If the element is zero than product is zero too.
 = 4 * ( -1) 3 + 1 * ( -8) * 3 6 = 10 5
 = - 32 * 3 6 = 10 5
= - 32 * ( 3 * 5 - 6 * 10 ) =
= - 32 * ( 15 - 60 ) =
= 1440