Example of Finding the Determinant of a 5x5 Matrix

This solution was made using 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}

-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}

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}

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}

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}

-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}

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}

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}

		-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}

		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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