Cramers Rule: A Step-by-Step Guide to Solving Linear Equations

Solving a system of linear equations using Cramers Rule involves calculating the determinant of the coefficient matrix and replacing columns with the constants to find the determinants for each variable. This method provides a straightforward way to find solutions when the number of equations equals the number of unknowns and the determinant is non-zero. It's especially useful for smaller systems where determinants can be computed easily.

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1. Introduction to the Cramer's Rule:

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with the help of determinants. Named after Gabriel Cramer, this rule provides a direct method to find the values of unknowns in a linear system, assuming that the determinant of the coefficient matrix is non-zero.

2. What is Cramer's Rule:

Cramer's Rule is a formula for solving linear equations with as many equations as unknowns. It uses determinants to express the solutions explicitly. Each variable in the system is found by dividing the determinant of a matrix (formed by replacing one column of the coefficient matrix with the constants) by the determinant of the coefficient matrix.

3. How to solve a system of equations by Cramer's Rule:

To solve a system of equations using Cramer's Rule, follow these steps:

• Write the system of equations in matrix form AX = B.
• Multiply corresponding elements and sum them up to get the elements of the resultant matrix.
• Replace each column of A with the column vector B to form new matrices.
• Calculate the determinants of these new matrices.
• Solve for each variable using the formula $x_i = \frac{det(A_i)}{det(A)}$

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4. Rules for Cramer's Rule:

• The system must have the same number of equations and unknowns.
• The determinant of the coefficient matrix A must be non-zero.
• Each variable is solved by substituting the column of constants into the coefficient matrix and calculating the determinant.

5. Cramer's Rule Formula for 2x2:

For a 2x2 system of Linear system of equations: $\begin{cases} a_1x b_1y = c_1 \\ a_2x b_2y = c_2 \end{cases}$

The solution of the above system of equations is:
x = $\frac{D_x}{D}$ and y = $\frac{D_y}{D}$
where,
$D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}$, $D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}$ and $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$

6. Cramer's Rule Formula for 3x3

For a 2x2 system of Linear system of equations: $\begin{cases} a_1x b_1y c_1z = d_1 \\ a_2x b_2y c_2z = d_2 \\ a_3x b_3y c_3z = d_3 \end{cases}$

The solution of the above system of equations is:
x = $\frac{D_x}{D}$, y = $\frac{D_y}{D}$, and z = $\frac{D_z}{D}$
where,
$D_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \\ \end{vmatrix}$, $D_y = \begin{vmatrix} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \\ \end{vmatrix}$, $D_z = \begin{vmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \\ \end{vmatrix}$, and $D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}$

7. Properties of Cramer's Rule:

8. Cramer's Rule Solved Examples:

Example 1: Solving a 2x2 System $\begin{cases} 3x 2y = 16 \\ x - 4y = -2 \end{cases}$

Solution: Here, the determinant of the coefficient matrix is zero, indicating that the system does not have a unique solution.

Example 2: Solving a 3x3 System $\begin{cases} x y z = 6 \\ 2x - y 3z = 14 \\ 3x 4y 2z = 21 \end{cases}$

Solution: Using Cramer's Rule, we find:
$D_x = \begin{vmatrix} 6 & 1 & 1 \\ 14 & -1 & 3 \\ 21 & 4 & 2 \\ \end{vmatrix} = 28$, $D_y = \begin{vmatrix} 1 & 6 & 1 \\ 2 & 14 & 3 \\ 3 & 21 & 2 \\ \end{vmatrix} = -5$, $D_z = \begin{vmatrix} 1 & 1 & 6 \\ 2 & -1 & 14 \\ 3 & 4 & 21 \\ \end{vmatrix} = -11$, and $D = \begin{vmatrix} 1 & 1 & 1 \\ 2 & -1 & 3 \\ 3 & 4 & 2 \\ \end{vmatrix} = 2$

Therefore, x = 14, y = $\frac{-5}{2}$, and z = $\frac{-11}{2}$

9. Practice questions on Cramer's Rule:

Question 1: Solve the system of equation: $\begin{cases} 3x 2y = 16 \\ x - 4y = -2 \end{cases}$

Question 2: Solve the system of equation: $\begin{cases} x 2y 3z = 1 \\ 2x - y z = 4 \\ 3x y - z = 3 \end{cases}$

10. FAQs on Cramer's Rule:

What is Cramer's Rule for 2x2?

Cramer's Rule for a 2x2 system provides the solution by using the determinants of matrices obtained by replacing columns of the coefficient matrix with the constant terms.

How does Cramer's Rule work?

Cramer's Rule works by solving each variable in a system of linear equations as the ratio of two determinants: the determinant of a matrix formed by replacing a column with the constants and the determinant of the coefficient matrix.

Define Cramer's Rule.

Cramer's Rule is a theorem in linear algebra that gives a unique solution to a system of linear equations with as many equations as unknowns, using determinants.

Who invented Cramer's Rule for a system of equations?

Cramer's Rule was named after Gabriel Cramer, a Swiss mathematician who introduced it in 1750.

Does Cramer's Rule apply when the determinant is zero?

No, Cramer's Rule does not apply if the determinant of the coefficient matrix is zero, as this indicates that the system does not have a unique solution.

What are $D_x$ and $D_y$ in Cramer's Rule formula?

$D_x$ and $D_y$ are the determinants of matrices obtained by replacing the x and y columns of the coefficient matrix with the constants, respectively.

What are the limitations of Cramer's Rule?

Cramer's Rule is computationally expensive for large systems and does not apply if the determinant of the coefficient matrix is zero.

11. Real-life application of Cramer's Rule:

Cramer's Rule is a powerful mathematical tool used to solve systems of linear equations using determinants. Named after the Swiss mathematician Gabriel Cramer, this rule offers a straightforward method to find solutions for equations in the form of matrices. It's particularly useful in cases where the system is small and the determinants are easy to calculate.

12. Conclusion:

Cramer's Rule is a powerful yet straightforward method for solving systems of linear equations using determinants. It is highly effective for small systems where the determinant of the coefficient matrix is non-zero. While its practical use diminishes with larger systems due to computational complexity, it remains a valuable tool in fields such as engineering and physics. Understanding its application and limitations can greatly enhance one's problem-solving skills in linear algebra.

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