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 =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

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

-3	-3	-4 + ( -3) * ( -1)
-4	4	-6 + ( -4) * ( -1)
4	-6	5 + 4 * ( -1)

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

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

The elements of column 3 multiplied by -2 are added to the corresponding elements of column 1. more info

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

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

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

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

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

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

=	-1	-5	-1	=
	0	0	-2
	2	-4	1

Expand the determinant along the row 2. more info

-1	-5	-1
0	0	-2
2	-4	1

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

		-5	-1
		-4	1

-1	-5	-1
0	0	-2
2	-4	1

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		-1	-1
		2	1

-1	-5	-1
0	0	-2
2	-4	1

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

-2

		-1	-5
		2	-4

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

= ( -1) ^{2 + 3} * ( -2) *		-1	-5		=
		2	-4

= 2 *		-1	-5		=
		2	-4

= 2 * ( -1 * ( -4) - ( -5) * 2 ) =

= 2 * ( 4 + 10 ) =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the row 1. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 1

Element

Row 1 and column 1
have been deleted

( -1) ^{1 + 1}

-3

		4	-6
		-6	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 2

Element

Row 1 and column 2
have been deleted

( -1) ^{1 + 2}

-3

		-4	-6
		4	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

-4

		-4	4
		4	-6

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

= ( -1) ^{1 + 1} * ( -3) *		4	-6
		-6	5

+ ( -1) ^{1 + 2} * ( -3) *		-4	-6
		4	5

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

= - 3 *		4	-6
		-6	5

+ 3 *		-4	-6
		4	5

- 4 *		-4	4		=
		4	-6

= - 3 * ( 4 * 5 - ( -6) * ( -6) )

+ 3 * ( -4 * 5 - ( -6) * 4 )

- 4 * ( -4 * ( -6) - 4 * 4 ) =

= - 3 * ( 20 - 36 )

+ 3 * ( -20 + 24 )

- 4 * ( 24 - 16 ) =

= 48

+ 12

- 32 =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the row 2. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

-4

		-3	-4
		-6	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		-3	-4
		4	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

-6

		-3	-3
		4	-6

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

= ( -1) ^{2 + 1} * ( -4) *		-3	-4
		-6	5

+ ( -1) ^{2 + 2} * 4 *		-3	-4
		4	5

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

= 4 *		-3	-4
		-6	5

+ 4 *		-3	-4
		4	5

+ 6 *		-3	-3		=
		4	-6

= 4 * ( -3 * 5 - ( -4) * ( -6) )

+ 4 * ( -3 * 5 - ( -4) * 4 )

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

= 4 * ( -15 - 24 )

+ 4 * ( -15 + 16 )

+ 6 * ( 18 + 12 ) =

= -156

+ 4

+ 180 =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the row 3. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

		-3	-4
		4	-6

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

-6

		-3	-4
		-4	-6

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		-3	-3
		-4	4

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

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

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

+ ( -1) ^{3 + 3} * 5 *		-3	-3		=
		-4	4

= 4 *		-3	-4
		4	-6

+ 6 *		-3	-4
		-4	-6

+ 5 *		-3	-3		=
		-4	4

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

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

+ 5 * ( -3 * 4 - ( -3) * ( -4) ) =

= 4 * ( 18 + 16 )

+ 6 * ( 18 - 16 )

+ 5 * ( -12 - 12 ) =

= 136

+ 12

- 120 =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the column 1. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 1

Element

Row 1 and column 1
have been deleted

( -1) ^{1 + 1}

-3

		4	-6
		-6	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 1

Element

Row 2 and column 1
have been deleted

( -1) ^{2 + 1}

-4

		-3	-4
		-6	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

		-3	-4
		4	-6

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

= ( -1) ^{1 + 1} * ( -3) *		4	-6
		-6	5

+ ( -1) ^{2 + 1} * ( -4) *		-3	-4
		-6	5

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

= - 3 *		4	-6
		-6	5

+ 4 *		-3	-4
		-6	5

+ 4 *		-3	-4		=
		4	-6

= - 3 * ( 4 * 5 - ( -6) * ( -6) )

+ 4 * ( -3 * 5 - ( -4) * ( -6) )

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

= - 3 * ( 20 - 36 )

+ 4 * ( -15 - 24 )

+ 4 * ( 18 + 16 ) =

= 48

- 156

+ 136 =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the column 2. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 2

Element

Row 1 and column 2
have been deleted

( -1) ^{1 + 2}

-3

		-4	-6
		4	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 2

Element

Row 2 and column 2
have been deleted

( -1) ^{2 + 2}

		-3	-4
		4	5

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

-6

		-3	-4
		-4	-6

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

= ( -1) ^{1 + 2} * ( -3) *		-4	-6
		4	5

+ ( -1) ^{2 + 2} * 4 *		-3	-4
		4	5

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

= 3 *		-4	-6
		4	5

+ 4 *		-3	-4
		4	5

+ 6 *		-3	-4		=
		-4	-6

= 3 * ( -4 * 5 - ( -6) * 4 )

+ 4 * ( -3 * 5 - ( -4) * 4 )

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

= 3 * ( -20 + 24 )

+ 4 * ( -15 + 16 )

+ 6 * ( 18 - 16 ) =

= 12

+ 4

+ 12 =

= 28

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

det A =	-3	-3	-4	=
	-4	4	-6
	4	-6	5

Expand the determinant along the column 3. more info

-3	-3	-4
-4	4	-6
4	-6	5

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

-4

		-4	4
		4	-6

-3	-3	-4
-4	4	-6
4	-6	5

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

-6

		-3	-3
		4	-6

-3	-3	-4
-4	4	-6
4	-6	5

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		-3	-3
		-4	4

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

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

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

+ ( -1) ^{3 + 3} * 5 *		-3	-3		=
		-4	4

= - 4 *		-4	4
		4	-6

+ 6 *		-3	-3
		4	-6

+ 5 *		-3	-3		=
		-4	4

= - 4 * ( -4 * ( -6) - 4 * 4 )

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

+ 5 * ( -3 * 4 - ( -3) * ( -4) ) =

= - 4 * ( 24 - 16 )

+ 6 * ( 18 + 12 )

+ 5 * ( -12 - 12 ) =

= -32

+ 180

- 120 =

= 28

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