Example №2. Solving of a 3x3 System of Linear Equations by the Cramer's RuleThis solution was made using the calculator presented on the site. It is necessary to solve the system of linear equations using Cramer's rule.
Let's write the Cramer's rule: x_{1} = det A_{1} / det A x_{2} = det A_{2} / det A x_{3} = det A_{3} / 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.
The elements of row 1 are added to the corresponding elements of row 3. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the row 3. more info
Products are summed. If the element is zero than product is zero too.
= 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 A_{1}. more info It is necessary to change column 1 in determinant A to the column of the right side of the system.
The elements of column 2 are added to the corresponding elements of column 3. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the column 3. more info
Products are summed. If the element is zero than product is zero too.
= 8 * ( 3)  2 * 18 = = 24  36 = = 12 Let's calculate the determinant A_{2}. more info It is necessary to change column 2 in determinant A to the column of the right side of the system.
The elements of column 1 are added to the corresponding elements of column 3. more info
This elementary transformation does not change the value of the determinant.
The elements of row 1 multiplied by 2 are added to the corresponding elements of row 3. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the column 3. more info
Products are summed. If the element is zero than product is zero too.
=  ( 3 * ( 11)  18 * 1 ) = =  ( 33  18 ) = = 15 Let's calculate the determinant A_{3}. more info It is necessary to change column 3 in determinant A to the column of the right side of the system.
The elements of row 1 are added to the corresponding elements of row 3. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the row 3. more info
Products are summed. If the element is zero than product is zero too.
=  3 * ( 1 * ( 3)  2 * ( 3) ) = =  3 * ( 3 + 6 ) = = 9 Result: x_{1} = det A_{1} / det A = 12/3 = 4 x_{2} = det A_{2} / det A = 15/3 = 5 x_{3} = det A_{3} / det A = 9/3 = 3
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