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 = 1 1 -1 -1 =
2 18 -2 2
3 3 18 -3
4 5 6 -4
The elements of row 1 multiplied by -3 are added to the corresponding elements of row 3.   more info
1 1 -1 -1
2 18 -2 2
3 + 1 * ( -3) 3 + 1 * ( -3) 18 + ( -1) * ( -3) -3 + ( -1) * ( -3)
4 5 6 -4
This elementary transformation does not change the value of the determinant.
= 1 1 -1 -1 =
2 18 -2 2
0 0 21 0
4 5 6 -4
Expand the determinant along the row 3.   more info
1 1 -1 -1
2 18 -2 2
0 0 21 0
4 5 6 -4
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
1 -1 -1
18 -2 2
5 6 -4
1 1 -1 -1
2 18 -2 2
0 0 21 0
4 5 6 -4
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
1 -1 -1
2 -2 2
4 6 -4
1 1 -1 -1
2 18 -2 2
0 0 21 0
4 5 6 -4
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 21 *
1 1 -1
2 18 2
4 5 -4
1 1 -1 -1
2 18 -2 2
0 0 21 0
4 5 6 -4
Row number 3
Column number 4
Element Row 3 and column 4
have been deleted
( -1) 3 + 4 * 0 *
1 1 -1
2 18 -2
4 5 6
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 3 * 21 * 1 1 -1 =
2 18 2
4 5 -4
= 21 * 1 1 -1 =
2 18 2
4 5 -4
The elements of row 1 multiplied by -4 are added to the corresponding elements of row 3.   more info
1 1 -1
2 18 2
4 + 1 * ( -4) 5 + 1 * ( -4) -4 + ( -1) * ( -4)
This elementary transformation does not change the value of the determinant.
= 21 * 1 1 -1 =
2 18 2
0 1 0
Expand the determinant along the row 3.   more info
1 1 -1
2 18 2
0 1 0
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
1 -1
18 2
1 1 -1
2 18 2
0 1 0
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 1 *
1 -1
2 2
1 1 -1
2 18 2
0 1 0
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
1 1
2 18
Products are summed. If the element is zero than product is zero too.
= 21 * ( -1) 3 + 2 * 1 * 1 -1 =
2 2
= - 21 * 1 -1 =
2 2
= - 21 * ( 1 * 2 - ( -1) * 2 ) =
= - 21 * ( 2 + 2 ) =
= -84
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