Example of Finding the Determinant of a 5x5 Matrix

This solution has been done by the calculator presented on the site.

Let's calculate the determinant A using a elementary transformations.
det A = 0 12 -12 12 6 =
-3 -9 9 9 -6
0 0 -2 4 -2
-3 -17 13 3 -8
0 0 4 -8 0
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 4.   more info
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
-3 + ( -3) * ( -1) -17 + ( -9) * ( -1) 13 + 9 * ( -1) 3 + 9 * ( -1) -8 + ( -6) * ( -1)
0 0 4 -8 0
This elementary transformation does not change the value of the determinant.
= 0 12 -12 12 6 =
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Expand the determinant along the column 1.   more info
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 0 *
-9 9 9 -6
0 -2 4 -2
-8 4 -6 -2
0 4 -8 0
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * -3 *
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 4 -8 0
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
12 -12 12 6
-9 9 9 -6
-8 4 -6 -2
0 4 -8 0
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Row number 4
Column number 1
Element Row 4 and column 1
have been deleted
( -1) 4 + 1 * 0 *
12 -12 12 6
-9 9 9 -6
0 -2 4 -2
0 4 -8 0
0 12 -12 12 6
-3 -9 9 9 -6
0 0 -2 4 -2
0 -8 4 -6 -2
0 0 4 -8 0
Row number 5
Column number 1
Element Row 5 and column 1
have been deleted
( -1) 5 + 1 * 0 *
12 -12 12 6
-9 9 9 -6
0 -2 4 -2
-8 4 -6 -2
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 1 * ( -3) * 12 -12 12 6 =
0 -2 4 -2
-8 4 -6 -2
0 4 -8 0
= 3 * 12 -12 12 6 =
0 -2 4 -2
-8 4 -6 -2
0 4 -8 0
The elements of row 2 multiplied by 2 are added to the corresponding elements of row 4.   more info
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 + 0 * 2 4 + ( -2) * 2 -8 + 4 * 2 0 + ( -2) * 2
This elementary transformation does not change the value of the determinant.
= 3 * 12 -12 12 6 =
0 -2 4 -2
-8 4 -6 -2
0 0 0 -4
Expand the determinant along the row 4.   more info
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 0 0 -4
Row number 4
Column number 1
Element Row 4 and column 1
have been deleted
( -1) 4 + 1 * 0 *
-12 12 6
-2 4 -2
4 -6 -2
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 0 0 -4
Row number 4
Column number 2
Element Row 4 and column 2
have been deleted
( -1) 4 + 2 * 0 *
12 12 6
0 4 -2
-8 -6 -2
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 0 0 -4
Row number 4
Column number 3
Element Row 4 and column 3
have been deleted
( -1) 4 + 3 * 0 *
12 -12 6
0 -2 -2
-8 4 -2
12 -12 12 6
0 -2 4 -2
-8 4 -6 -2
0 0 0 -4
Row number 4
Column number 4
Element Row 4 and column 4
have been deleted
( -1) 4 + 4 * -4 *
12 -12 12
0 -2 4
-8 4 -6
Products are summed. If the element is zero than product is zero too.
= 3 * ( -1) 4 + 4 * ( -4) * 12 -12 12 =
0 -2 4
-8 4 -6
= - 12 * 12 -12 12 =
0 -2 4
-8 4 -6
The elements of column 2 multiplied by 2 are added to the corresponding elements of column 3.   more info
12 -12 12 + ( -12) * 2
0 -2 4 + ( -2) * 2
-8 4 -6 + 4 * 2
This elementary transformation does not change the value of the determinant.
= -12 * 12 -12 -12 =
0 -2 0
-8 4 2
Expand the determinant along the row 2.   more info
12 -12 -12
0 -2 0
-8 4 2
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 0 *
-12 -12
4 2
12 -12 -12
0 -2 0
-8 4 2
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * -2 *
12 -12
-8 2
12 -12 -12
0 -2 0
-8 4 2
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 0 *
12 -12
-8 4
Products are summed. If the element is zero than product is zero too.
= -12 * ( -1) 2 + 2 * ( -2) * 12 -12 =
-8 2
= 24 * 12 -12 =
-8 2
= 24 * ( 12 * 2 - ( -12) * ( -8) ) =
= 24 * ( 24 - 96 ) =
= -1728
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