Example of Finding the Determinant of a 3x3 Matrix

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

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

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

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

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

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

=	1	-2	-4	=
	3	0	3
	-2	8	4

The elements of column 1 multiplied by -1 are added to the corresponding elements of column 3. more info

1	-2	-4 + 1 * ( -1)
3	0	3 + 3 * ( -1)
-2	8	4 + ( -2) * ( -1)

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

=	1	-2	-5	=
	3	0	0
	-2	8	6

Expand the determinant along the row 2. more info

1	-2	-5
3	0	0
-2	8	6

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

		-2	-5
		8	6

1	-2	-5
3	0	0
-2	8	6

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		1	-5
		-2	6

1	-2	-5
3	0	0
-2	8	6

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

		1	-2
		-2	8

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

= ( -1) ^{2 + 1} * 3 *		-2	-5		=
		8	6

= - 3 *		-2	-5		=
		8	6

= - 3 * ( -2 * 6 - ( -5) * 8 ) =

= - 3 * ( -12 + 40 ) =

= -84

2. Let's calculate the determinant A by expanding along the row 1.

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

Expand the determinant along the row 1. more info

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

Row number 1
Column number 1

Element

Row 1 and column 1
have been deleted

( -1) ^{1 + 1}

		3	3
		6	4

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

Row number 1
Column number 2

Element

Row 1 and column 2
have been deleted

( -1) ^{1 + 2}

-1

		3	3
		-2	4

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

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

-4

		3	3
		-2	6

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

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

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

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

=		3	3
		6	4

+		3	3
		-2	4

- 4 *		3	3		=
		-2	6

= ( 3 * 4 - 3 * 6 )

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

- 4 * ( 3 * 6 - 3 * ( -2) ) =

= ( 12 - 18 )

+ ( 12 + 6 )

- 4 * ( 18 + 6 ) =

= -6

+ 18

- 96 =

= -84

3. Let's calculate the determinant A by expanding along the row 2.

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

Expand the determinant along the row 2. more info

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

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

		-1	-4
		6	4

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

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		1	-4
		-2	4

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

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

		1	-1
		-2	6

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

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

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

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

= - 3 *		-1	-4
		6	4

+ 3 *		1	-4
		-2	4

- 3 *		1	-1		=
		-2	6

= - 3 * ( -1 * 4 - ( -4) * 6 )

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

- 3 * ( 1 * 6 - ( -1) * ( -2) ) =

= - 3 * ( -4 + 24 )

+ 3 * ( 4 - 8 )

- 3 * ( 6 - 2 ) =

= -60

- 12

- 12 =

= -84

4. Let's calculate the determinant A by expanding along the row 3.

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

Expand the determinant along the row 3. more info

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

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

-2

		-1	-4
		3	3

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

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

		1	-4
		3	3

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

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		1	-1
		3	3

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

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

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

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

= - 2 *		-1	-4
		3	3

- 6 *		1	-4
		3	3

+ 4 *		1	-1		=
		3	3

= - 2 * ( -1 * 3 - ( -4) * 3 )

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

+ 4 * ( 1 * 3 - ( -1) * 3 ) =

= - 2 * ( -3 + 12 )

- 6 * ( 3 + 12 )

+ 4 * ( 3 + 3 ) =

= -18

- 90

+ 24 =

= -84

5. Let's calculate the determinant A by expanding along the column 1.

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

Expand the determinant along the column 1. more info

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

Row number 1
Column number 1

Element

Row 1 and column 1
have been deleted

( -1) ^{1 + 1}

		3	3
		6	4

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

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

		-1	-4
		6	4

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

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

-2

		-1	-4
		3	3

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

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

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

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

=		3	3
		6	4

- 3 *		-1	-4
		6	4

- 2 *		-1	-4		=
		3	3

= ( 3 * 4 - 3 * 6 )

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

- 2 * ( -1 * 3 - ( -4) * 3 ) =

= ( 12 - 18 )

- 3 * ( -4 + 24 )

- 2 * ( -3 + 12 ) =

= -6

- 60

- 18 =

= -84

6. Let's calculate the determinant A by expanding along the column 2.

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

Expand the determinant along the column 2. more info

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

Row number 1
Column number 2

Element

Row 1 and column 2
have been deleted

( -1) ^{1 + 2}

-1

		3	3
		-2	4

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

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		1	-4
		-2	4

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

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

		1	-4
		3	3

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

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

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

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

=		3	3
		-2	4

+ 3 *		1	-4
		-2	4

- 6 *		1	-4		=
		3	3

= ( 3 * 4 - 3 * ( -2) )

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

- 6 * ( 1 * 3 - ( -4) * 3 ) =

= ( 12 + 6 )

+ 3 * ( 4 - 8 )

- 6 * ( 3 + 12 ) =

= 18

- 12

- 90 =

= -84

7. Let's calculate the determinant A by expanding along the column 3.

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

Expand the determinant along the column 3. more info

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

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

-4

		3	3
		-2	6

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

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

		1	-1
		-2	6

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

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		1	-1
		3	3

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

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

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

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

= - 4 *		3	3
		-2	6

- 3 *		1	-1
		-2	6

+ 4 *		1	-1		=
		3	3

= - 4 * ( 3 * 6 - 3 * ( -2) )

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

+ 4 * ( 1 * 3 - ( -1) * 3 ) =

= - 4 * ( 18 + 6 )

- 3 * ( 6 - 2 )

+ 4 * ( 3 + 3 ) =

= -96

- 12

+ 24 =

= -84

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