Example of Finding the Determinant of a 4x4 Matrix

This solution was made using 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}

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}

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}

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}

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}

		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
		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}

		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

Please do not forget to support the site with a link.