Example of Finding the Determinant of a 4x4 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 = 3 -3 -5 8 =
-3 2 4 -6
2 -5 -7 5
-4 3 5 -6
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 4.   more info
3 -3 -5 8
-3 2 4 -6
2 -5 -7 5
-4 + ( -3) * ( -1) 3 + 2 * ( -1) 5 + 4 * ( -1) -6 + ( -6) * ( -1)
This elementary transformation does not change the value of the determinant.
= 3 -3 -5 8 =
-3 2 4 -6
2 -5 -7 5
-1 1 1 0
The elements of column 1 are added to the corresponding elements of column 2.   more info
3 -3 + 3 -5 8
-3 2 + ( -3) 4 -6
2 -5 + 2 -7 5
-1 1 + ( -1) 1 0
This elementary transformation does not change the value of the determinant.
= 3 0 -5 8 =
-3 -1 4 -6
2 -3 -7 5
-1 0 1 0
The elements of column 1 are added to the corresponding elements of column 3.   more info
3 0 -5 + 3 8
-3 -1 4 + ( -3) -6
2 -3 -7 + 2 5
-1 0 1 + ( -1) 0
This elementary transformation does not change the value of the determinant.
= 3 0 -2 8 =
-3 -1 1 -6
2 -3 -5 5
-1 0 0 0
Expand the determinant along the row 4.   more info
3 0 -2 8
-3 -1 1 -6
2 -3 -5 5
-1 0 0 0
Row number 4
Column number 1
Element Row 4 and column 1
have been deleted
( -1) 4 + 1 * -1 *
0 -2 8
-1 1 -6
-3 -5 5
3 0 -2 8
-3 -1 1 -6
2 -3 -5 5
-1 0 0 0
Row number 4
Column number 2
Element Row 4 and column 2
have been deleted
( -1) 4 + 2 * 0 *
3 -2 8
-3 1 -6
2 -5 5
3 0 -2 8
-3 -1 1 -6
2 -3 -5 5
-1 0 0 0
Row number 4
Column number 3
Element Row 4 and column 3
have been deleted
( -1) 4 + 3 * 0 *
3 0 8
-3 -1 -6
2 -3 5
3 0 -2 8
-3 -1 1 -6
2 -3 -5 5
-1 0 0 0
Row number 4
Column number 4
Element Row 4 and column 4
have been deleted
( -1) 4 + 4 * 0 *
3 0 -2
-3 -1 1
2 -3 -5
Products are summed. If the element is zero than product is zero too.
= ( -1) 4 + 1 * ( -1) * 0 -2 8 =
-1 1 -6
-3 -5 5
= 0 -2 8 =
-1 1 -6
-3 -5 5
The elements of row 2 multiplied by -3 are added to the corresponding elements of row 3.   more info
0 -2 8
-1 1 -6
-3 + ( -1) * ( -3) -5 + 1 * ( -3) 5 + ( -6) * ( -3)
This elementary transformation does not change the value of the determinant.
= 0 -2 8 =
-1 1 -6
0 -8 23
Expand the determinant along the column 1.   more info
0 -2 8
-1 1 -6
0 -8 23
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 0 *
1 -6
-8 23
0 -2 8
-1 1 -6
0 -8 23
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * -1 *
-2 8
-8 23
0 -2 8
-1 1 -6
0 -8 23
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
-2 8
1 -6
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 1 * ( -1) * -2 8 =
-8 23
= -2 8 =
-8 23
= -2 * 23 - 8 * ( -8) =
= -46 + 64 =
= 18
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