Example of Solving a System of Linear Equations by Cramer's Rule.

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It is necessary to solve the system of linear equations using Cramer's rule.
Знак системы-2 x1+x2+x3 = -13
-x1+x2+2x3 = -9
3x1+x2+x3 = 12
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.
Знак системы-2 x1+x2+x3 = -13
-x1+x2+2x3 = -9
3x1+x2+x3 = 12
det A = -2 1 1 =
-1 1 2
3 1 1
The elements of row 1 multiplied by -1 are added to the corresponding elements of row 3.   more info
-2 1 1
-1 1 2
3 + ( -2) * ( -1) 1 + 1 * ( -1) 1 + 1 * ( -1)
This elementary transformation does not change the value of the determinant.
= -2 1 1 =
-1 1 2
5 0 0
Expand the determinant along the row 3.   more info
-2 1 1
-1 1 2
5 0 0
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 5 *
1 1
1 2
-2 1 1
-1 1 2
5 0 0
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
-2 1
-1 2
-2 1 1
-1 1 2
5 0 0
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
-2 1
-1 1
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 1 * 5 * 1 1 =
1 2
= 5 * 1 1 =
1 2
= 5 * ( 1 * 2 - 1 * 1 ) =
= 5 * ( 2 - 1 ) =
= 5
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
Знак системы-2 x1+x2+x3 = -13
-x1+x2+2x3 = -9
3x1+x2+x3 = 12
-2 1 1
-1 1 2
3 1 1
-13 1 1
-9 1 2
12 1 1
det A1 = -13 1 1 =
-9 1 2
12 1 1
The elements of row 1 multiplied by -1 are added to the corresponding elements of row 3.   more info
-13 1 1
-9 1 2
12 + ( -13) * ( -1) 1 + 1 * ( -1) 1 + 1 * ( -1)
This elementary transformation does not change the value of the determinant.
= -13 1 1 =
-9 1 2
25 0 0
Expand the determinant along the row 3.   more info
-13 1 1
-9 1 2
25 0 0
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 25 *
1 1
1 2
-13 1 1
-9 1 2
25 0 0
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
-13 1
-9 2
-13 1 1
-9 1 2
25 0 0
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
-13 1
-9 1
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 1 * 25 * 1 1 =
1 2
= 25 * 1 1 =
1 2
= 25 * ( 1 * 2 - 1 * 1 ) =
= 25 * ( 2 - 1 ) =
= 25
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
Знак системы-2 x1+x2+x3 = -13
-x1+x2+2x3 = -9
3x1+x2+x3 = 12
-2 1 1
-1 1 2
3 1 1
-2 -13 1
-1 -9 2
3 12 1
det A2 = -2 -13 1 =
-1 -9 2
3 12 1
The elements of row 1 are added to the corresponding elements of row 3.   more info
-2 -13 1
-1 -9 2
3 + ( -2) 12 + ( -13) 1 + 1
This elementary transformation does not change the value of the determinant.
= -2 -13 1 =
-1 -9 2
1 -1 2
The elements of row 3 multiplied by 2 are added to the corresponding elements of row 1.   more info
-2 + 1 * 2 -13 + ( -1) * 2 1 + 2 * 2
-1 -9 2
1 -1 2
This elementary transformation does not change the value of the determinant.
= 0 -15 5 =
-1 -9 2
1 -1 2
The elements of row 3 are added to the corresponding elements of row 2.   more info
0 -15 5
-1 + 1 -9 + ( -1) 2 + 2
1 -1 2
This elementary transformation does not change the value of the determinant.
= 0 -15 5 =
0 -10 4
1 -1 2
Expand the determinant along the column 1.   more info
0 -15 5
0 -10 4
1 -1 2
Row number 1
Column number 1
Element Row 1 and column 1
have been deleted
( -1) 1 + 1 * 0 *
-10 4
-1 2
0 -15 5
0 -10 4
1 -1 2
Row number 2
Column number 1
Element Row 2 and column 1
have been deleted
( -1) 2 + 1 * 0 *
-15 5
-1 2
0 -15 5
0 -10 4
1 -1 2
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 1 *
-15 5
-10 4
Products are summed. If the element is zero than product is zero too.
= ( -1) 3 + 1 * 1 * -15 5 =
-10 4
= -15 5 =
-10 4
= -15 * 4 - 5 * ( -10) =
= -60 + 50 =
= -10
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
Знак системы-2 x1+x2+x3 = -13
-x1+x2+2x3 = -9
3x1+x2+x3 = 12
-2 1 1
-1 1 2
3 1 1
-2 1 -13
-1 1 -9
3 1 12
det A3 = -2 1 -13 =
-1 1 -9
3 1 12
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 1.   more info
-2 + ( -1) * ( -1) 1 + 1 * ( -1) -13 + ( -9) * ( -1)
-1 1 -9
3 1 12
This elementary transformation does not change the value of the determinant.
= -1 0 -4 =
-1 1 -9
3 1 12
The elements of row 2 multiplied by -1 are added to the corresponding elements of row 3.   more info
-1 0 -4
-1 1 -9
3 + ( -1) * ( -1) 1 + 1 * ( -1) 12 + ( -9) * ( -1)
This elementary transformation does not change the value of the determinant.
= -1 0 -4 =
-1 1 -9
4 0 21
Expand the determinant along the column 2.   more info
-1 0 -4
-1 1 -9
4 0 21
Row number 1
Column number 2
Element Row 1 and column 2
have been deleted
( -1) 1 + 2 * 0 *
-1 -9
4 21
-1 0 -4
-1 1 -9
4 0 21
Row number 2
Column number 2
Element Row 2 and column 2
have been deleted
( -1) 2 + 2 * 1 *
-1 -4
4 21
-1 0 -4
-1 1 -9
4 0 21
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
-1 -4
-1 -9
Products are summed. If the element is zero than product is zero too.
= ( -1) 2 + 2 * 1 * -1 -4 =
4 21
= -1 -4 =
4 21
= -1 * 21 - ( -4) * 4 =
= -21 + 16 =
= -5
Result:
x1 = det A1 / det A = 25/5 = 5
x2 = det A2 / det A = -10/5 = -2
x3 = det A3 / det A = -5/5 = -1
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