Calculator of the Determinant of a 4x4 Matrix

Example №2. Finding the Determinant of a 4x4 Matrix

This solution was made using the calculator presented on the site.

Example №1. Finding the determinant of a 3x3 matrix
Example №3. Finding the determinant of a 5x5 matrix

Let's calculate the determinant A using a elementary transformations.

det A =	4	6	-2	4	=
	1	2	-3	1
	4	-2	1	0
	6	4	4	6

The elements of column 1 multiplied by -1 are added to the corresponding elements of column 4. more info

4	6	-2	4 + 4 * ( -1)
1	2	-3	1 + 1 * ( -1)
4	-2	1	0 + 4 * ( -1)
6	4	4	6 + 6 * ( -1)

This elementary transformation does not change the value of the determinant.

=	4	6	-2	0	=
	1	2	-3	0
	4	-2	1	-4
	6	4	4	0

Expand the determinant along the column 4. more info

4	6	-2	0
1	2	-3	0
4	-2	1	-4
6	4	4	0

Row number 1
Column number 4

Element

Row 1 and column 4
have been deleted

( -1) ^{1 + 4}

1	2	-3
4	-2	1
6	4	4

4	6	-2	0
1	2	-3	0
4	-2	1	-4
6	4	4	0

Row number 2
Column number 4

Element

Row 2 and column 4
have been deleted

( -1) ^{2 + 4}

4	6	-2
4	-2	1
6	4	4

4	6	-2	0
1	2	-3	0
4	-2	1	-4
6	4	4	0

Row number 3
Column number 4

Element

Row 3 and column 4
have been deleted

( -1) ^{3 + 4}

-4

4	6	-2
1	2	-3
6	4	4

4	6	-2	0
1	2	-3	0
4	-2	1	-4
6	4	4	0

Row number 4
Column number 4

Element

Row 4 and column 4
have been deleted

( -1) ^{4 + 4}

4	6	-2
1	2	-3
4	-2	1

Products are summed. If the element is zero then product is zero too.

= ( -1) ^{3 + 4} * ( -4) *	4	6	-2	=
	1	2	-3
	6	4	4

= 4 *	4	6	-2	=
	1	2	-3
	6	4	4

The elements of column 1 multiplied by -1 are added to the corresponding elements of column 2. more info

4	6 + 4 * ( -1)	-2
1	2 + 1 * ( -1)	-3
6	4 + 6 * ( -1)	4

This elementary transformation does not change the value of the determinant.

= 4 *	4	2	-2	=
	1	1	-3
	6	-2	4

The elements of column 2 multiplied by -2 are added to the corresponding elements of column 1. more info

4 + 2 * ( -2)	2	-2
1 + 1 * ( -2)	1	-3
6 + ( -2) * ( -2)	-2	4

This elementary transformation does not change the value of the determinant.

= 4 *	0	2	-2	=
	-1	1	-3
	10	-2	4

The elements of column 2 are added to the corresponding elements of column 3. more info

0	2	-2 + 2
-1	1	-3 + 1
10	-2	4 + ( -2)

This elementary transformation does not change the value of the determinant.

= 4 *	0	2	0	=
	-1	1	-2
	10	-2	2

Expand the determinant along the row 1. more info

0	2	0
-1	1	-2
10	-2	2

Row number 1
Column number 1

Element

Row 1 and column 1
have been deleted

( -1) ^{1 + 1}

		1	-2
		-2	2

0	2	0
-1	1	-2
10	-2	2

Row number 1
Column number 2

Element

Row 1 and column 2
have been deleted

( -1) ^{1 + 2}

		-1	-2
		10	2

0	2	0
-1	1	-2
10	-2	2

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

		-1	1
		10	-2

Products are summed. If the element is zero then product is zero too.

= 4 * ( -1) ^{1 + 2} * 2 *		-1	-2		=
		10	2

= - 8 *		-1	-2		=
		10	2

= - 8 * ( -1 * 2 - ( -2) * 10 ) =

= - 8 * ( -2 + 20 ) =

= -144

Go to the solution of your problem

Service for Solving Linear Programming Problems

Example №2. Finding the Determinant of a 4x4 Matrix