Example of Finding the Determinant of a 5x5 Matrix

This solution was 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
Expand the determinant along the column 5.   more info
-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
The elements of row 1 are added to the corresponding elements of row 3.   more info
-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
The elements of column 3 are added to the corresponding elements of column 4.   more info
-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
Expand the determinant along the row 3.   more info
-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
Expand the determinant along the row 3.   more info
-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


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