Calculator of Cramer's Rule 3x3

Example №2. Solving of a 3x3 System of Linear Equations by the Cramer's Rule

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

Example №1. Solving of a 2x2 system of linear equations by the Cramer's rule

It is necessary to solve the system of linear equations using Cramer's rule.

		x₁	+	2	x₂	-	2	x₃	=	-8
-	3	x₁	-	3	x₂	+	3	x₃	=	18
	-	x₁	-	2	x₂	+	3	x₃	=	5

Let's write the Cramer's rule:

x₁ = det A₁ / det A

x₂ = det A₂ / det A

x₃ = det A₃ / det A

It is impossible to divide by zero. Therefore, if the determinant of A is zero, then it is impossible to use Cramer's rule.

Let's calculate the determinant A. more info

The determinant A consists of the coefficients of the left side of the system.

		x₁	+	2	x₂	-	2	x₃	=	-8
-	3	x₁	-	3	x₂	+	3	x₃	=	18
	-	x₁	-	2	x₂	+	3	x₃	=	5

det A =	1	2	-2	=
	-3	-3	3
	-1	-2	3

The elements of row 1 are added to the corresponding elements of row 3. more info

1	2	-2
-3	-3	3
-1 + 1	-2 + 2	3 + ( -2)

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

=	1	2	-2	=
	-3	-3	3
	0	0	1

Expand the determinant along the row 3. more info

1	2	-2
-3	-3	3
0	0	1

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

		2	-2
		-3	3

1	2	-2
-3	-3	3
0	0	1

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

		1	-2
		-3	3

1	2	-2
-3	-3	3
0	0	1

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		1	2
		-3	-3

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

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

=		1	2		=
		-3	-3

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

= -3 + 6 =

= 3

The determinant A is not zero. It is possible to use the Cramer's rule.

Let's calculate the determinant A₁. more info

It is necessary to change column 1 in determinant A to the column of the right side of the system.

System

det A

det A₁

		x₁	+	2	x₂	-	2	x₃	=	-8
-	3	x₁	-	3	x₂	+	3	x₃	=	18
	-	x₁	-	2	x₂	+	3	x₃	=	5

1	2	-2
-3	-3	3
-1	-2	3

-8	2	-2
18	-3	3
5	-2	3

det A₁ =	-8	2	-2	=
	18	-3	3
	5	-2	3

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

-8	2	-2 + 2
18	-3	3 + ( -3)
5	-2	3 + ( -2)

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

=	-8	2	0	=
	18	-3	0
	5	-2	1

Expand the determinant along the column 3. more info

-8	2	0
18	-3	0
5	-2	1

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

		18	-3
		5	-2

-8	2	0
18	-3	0
5	-2	1

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

		-8	2
		5	-2

-8	2	0
18	-3	0
5	-2	1

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		-8	2
		18	-3

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

= ( -1) ^{3 + 3} * 1 *		-8	2		=
		18	-3

=		-8	2		=
		18	-3

= -8 * ( -3) - 2 * 18 =

= 24 - 36 =

= -12

Let's calculate the determinant A₂. more info

It is necessary to change column 2 in determinant A to the column of the right side of the system.

System

det A

det A₂

		x₁	+	2	x₂	-	2	x₃	=	-8
-	3	x₁	-	3	x₂	+	3	x₃	=	18
	-	x₁	-	2	x₂	+	3	x₃	=	5

1	2	-2
-3	-3	3
-1	-2	3

1	-8	-2
-3	18	3
-1	5	3

det A₂ =	1	-8	-2	=
	-3	18	3
	-1	5	3

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

1	-8	-2 + 1
-3	18	3 + ( -3)
-1	5	3 + ( -1)

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

=	1	-8	-1	=
	-3	18	0
	-1	5	2

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

1	-8	-1
-3	18	0
-1 + 1 * 2	5 + ( -8) * 2	2 + ( -1) * 2

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

=	1	-8	-1	=
	-3	18	0
	1	-11	0

Expand the determinant along the column 3. more info

1	-8	-1
-3	18	0
1	-11	0

Row number 1
Column number 3

Element

Row 1 and column 3
have been deleted

( -1) ^{1 + 3}

-1

		-3	18
		1	-11

1	-8	-1
-3	18	0
1	-11	0

Row number 2
Column number 3

Element

Row 2 and column 3
have been deleted

( -1) ^{2 + 3}

		1	-8
		1	-11

1	-8	-1
-3	18	0
1	-11	0

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

		1	-8
		-3	18

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

= ( -1) ^{1 + 3} * ( -1) *		-3	18		=
		1	-11

= -		-3	18		=
		1	-11

= - ( -3 * ( -11) - 18 * 1 ) =

= - ( 33 - 18 ) =

= -15

Let's calculate the determinant A₃. more info

It is necessary to change column 3 in determinant A to the column of the right side of the system.

System

det A

det A₃

		x₁	+	2	x₂	-	2	x₃	=	-8
-	3	x₁	-	3	x₂	+	3	x₃	=	18
	-	x₁	-	2	x₂	+	3	x₃	=	5

1	2	-2
-3	-3	3
-1	-2	3

1	2	-8
-3	-3	18
-1	-2	5

det A₃ =	1	2	-8	=
	-3	-3	18
	-1	-2	5

The elements of row 1 are added to the corresponding elements of row 3. more info

1	2	-8
-3	-3	18
-1 + 1	-2 + 2	5 + ( -8)

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

=	1	2	-8	=
	-3	-3	18
	0	0	-3

Expand the determinant along the row 3. more info

1	2	-8
-3	-3	18
0	0	-3

Row number 3
Column number 1

Element

Row 3 and column 1
have been deleted

( -1) ^{3 + 1}

		2	-8
		-3	18

1	2	-8
-3	-3	18
0	0	-3

Row number 3
Column number 2

Element

Row 3 and column 2
have been deleted

( -1) ^{3 + 2}

		1	-8
		-3	18

1	2	-8
-3	-3	18
0	0	-3

Row number 3
Column number 3

Element

Row 3 and column 3
have been deleted

( -1) ^{3 + 3}

-3

		1	2
		-3	-3

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

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

= - 3 *		1	2		=
		-3	-3

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

= - 3 * ( -3 + 6 ) =

= -9

Result:

x₁ = det A₁ / det A = -12/3 = -4

x₂ = det A₂ / det A = -15/3 = -5

x₃ = det A₃ / det A = -9/3 = -3

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Example №2. Solving of a 3x3 System of Linear Equations by the Cramer's Rule