Cramer's Rule

Properties of Cramer's rule are:
 

1. Cramer’s rule solves systems of linear equations using determinants.

2. Values of the variables can be determined by the determinants.

3. One determinant comes from the coefficients of variables (coefficient matrix).

4. Another determinant is created by replacing one column with the constants (right-hand side).

5. The value of a variable = determinant with replaced column ÷ coefficient determinant.

Let us see how Cramer’s rule works in the following system of equations: 

\(2x + y + z = 3\)
\(x - y - z = 0\)
\(x + 2y + z = 0\)

Write the given equations with all coefficients:

\(2x + 1y + 1z = 3\\ 1x - 1y - 1z = 0\\ 1x + 2y + 1z = 0\)

Create the coefficient matrix and identify its determinant (D): 

\(D= \begin{bmatrix} 2 & 1 & 1\\[0.3em] 1& -1 &-1 \\[0.3em] 1 & 2 &1 \end{bmatrix}\)

Here, the matrix only contains the coefficients of x, y and z.

The answer column:

\(\begin{bmatrix} 3\\[0.3em] 0 \\[0.3em] 0 \end{bmatrix}\)

Now, form D_x by replacing the 1^st column with the answer column:
 

D_{x = \( \begin{bmatrix} 3 & 1 & 1\\[0.3em] 0& -1 &-1 \\[0.3em] 0 & 2 &1 \end{bmatrix}\)}

Here, the determinant with x-values is replaced.

Create D_y and D_z by replacing the 2nd and 3rd columns with the answer column. 

D_y = \(\begin{bmatrix} 2 & 3 & 1\\[0.3em] 1& 0 &-1 \\[0.3em] 1& 0&1 \end{bmatrix}\)
 

D_{z = \(\begin{bmatrix} 2 & 1 & 3\\[0.3em] 1& -1&0 \\[0.3em] 1& 2&0\end{bmatrix}\)}

We can use cofactor expansion (3 × 3 rule). For a general 3 × 3 matrix, the formula is:

\(\begin{bmatrix} a& b& c\\[0.3em] d& e&f \\[0.3em] g& h&i\end{bmatrix} = aei +bfg + cdh - ceg -bdi - afh\)
 

D = \(\begin{bmatrix} 2 & 1 & 1\\[0.3em] 1& -1&-1 \\[0.3em] 1& 2&1\end{bmatrix}\)

D = Apply the cofactor expansion

\((2) (-1) (1) = -2\\ (1) (-1) (1) = -1\\ (1) (1) (2) = 2\\ (1) (-1) (1) = -1\\  (1) (-1) (1) = -1\\ (2) (1) (2) = 4  \)

\(D = (−2) + (−1) + 2 − (−1) − (−1) − 4 = −1 − 4 + 2 + 1 + 1 = -1\)

Hence, D = 3

Next \(D_x\)=   \(\begin{bmatrix} 3 & 1 & 1\\[0.3em] 0& -1&-1 \\[0.3em] 0& 2&1\end{bmatrix}\)

\(D_x\) = Apply the cofactor expansion

\((3) (-1) (1) = -3\\ (1) (-1) (0) = 0\\ (1) (0) (2) = 0\\ (1) (-1) (0) = 0\\ (1) (-1) (0) = 0\\ (3) (2) (1) = 6\)

\(D_x = −3 + 0 + 0 − 0 − 0 − 6 = −3 + 6 = 3\)

Thus, \(D_x = 3\)

\(D_y\) _{=  \(\begin{bmatrix} 2 & 3& 1\\[0.3em] 1& 0&-1 \\[0.3em] 1& 0&1\end{bmatrix}\)}

\(D_y\) ₌ Apply the cofactor expansion

\((2) (0) (1) = 0\\ (3) (-1) (1) = -3\\ (1) (1) (0) = 0\\ (1) (0) (1) = 0\\ (1) (-1) (2) = -2\\  (2) (0) (1) = 0 \)

\(D_y = 0 − 3 + 0 − 0 + 2 − 0 = −3 + 2 = −6\)

\(D_y = -6\)

\(D_z\) = \(\begin{bmatrix} 2 & 1& 3\\[0.3em] 1& -1&0 \\[0.3em] 1& 2&0\end{bmatrix}\)

\(D_z\) = Apply the cofactor expansion

\((2) (-1) (0) = 0\\ (1) (0) (1) = 0\\ (3) (1) (2) = 6\\ (3) (-1) (1) = -3\\ (2) (0) (2) = 0\\ (2) (1) (0) = 0\)

\(D_z = 0 + 0 + 6 − (−3) = 6 + 3 = 9\)

\(D_z = 9\)

Now, we can apply Cramer’s rule. 

\( x = \frac{D_x}{D} = \frac{3}{3} = 1 \)

\( y = \frac{D_y}{D} = \frac{-6}{3} = -2 \)

\( z = \frac{D_z}{D} = \frac{9}{3} = 3 \)

So, the values for the variables:

\(x = 1\\ y = -2\\ z = 3\)

Using Cramer’s rule, if a system has five equations in five unknowns, we can solve for just one variable if that is all we need. The system of equations consists of n variables x₁, x₂, x₃…..x_n can be written in the matrix form as:

AX = B

Here,
 

1. A = the square matrix that represents the coefficient matrix.
    
2. X = the column matrix with variables. 
    
3. B = the column matrix which contains the constants (present on the right side of the equations). 

For finding the value of x, we use the formula shown in the diagram. 

By using the above formula, we can determine the individual values of x, y, and z. 

There are specific conditions for Cramer’s rule. There are two important conditions for this rule, they are : 

First condition: It happens when D is equal to zero, and there is an infinite number of solutions. At least one determinant of the numerator, such as D_X, D_y, etc., is also 0.

Second condition: It happens when none of the numerators are zero, but D is equal to zero, and there is no solution.

When the determinant of the coefficient matrix (D) is not equal to zero, then Cramer’s rule is applicable. The system has a unique solution, and we can find a single value for each variable. If D is equal to zero, then there is an endless number of solutions or no solution for the system.

Using Cramer’s rule, let’s solve 2 systems of 2 equations with 2 variables. The equations are:
 

1. \( a_1 x + b_1 y = c_1\)
    
2. \( a_2 x + b_2 y = c_2\)

Writing the system in matrix form as:

AX = B 

Where A = the coefficient matrix

\( \begin{bmatrix} a_1 & b_1 & c_1\\[0.3em] d_2& e_2 & f_2 \\[0.3em] g_3 & h_3 &i_3\end{bmatrix}\)
 

X = the variable matrix 
 

\( \begin{bmatrix} x\\[0.3em] y \\[0.3em] z\end{bmatrix}\) 

B = The constant matrix 

\( \begin{bmatrix} d_1\\[0.3em] d_2 \\[0.3em] d_3\end{bmatrix}\)

Next, we can calculate the three determinants of the system.
 

• The determinant of the coefficient matrix A is D.
   
• To find the value of D_x, replace the first column of A with B.
   
• To find the value of D_y, the second column of A will be replaced with B.
   
• To find the value of D_z, replace the third column of A with B. 

The system has a unique solution when D is not equal to zero. 

\( x = \frac{D_x}{D} \)

\( y = \frac{D_y}{D} \)

\( z = \frac{D_z}{D}\)

Determinant calculation example:

Suppose a system is 

\(a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \)

Then the coefficient matrix A is 

\( \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \)

The determinant D is calculated as,

\( D = a_1 (b_2 c_3 - b_3 c_2) - b_1 (a_2 c_3 - a_3 c_2) + c_1 (a_2 b_3 - a_3 b_2) \)
 

Cramer's rule chart

When D ≠ 0, Cramer’s rule is applicable, then the system has a unique solution. The system has an infinite number of solutions when D = 0. There is no solution for the system when none of the numerators is 0. Here is a visual representation of Cramer’s rule, and it explains what type of solution it might have.

Here are some of the tips and tricks for students to master Cramer's rule and their parents to help their kids in learning:
 

1. Think of Cramer's rule as a symmetric way to solve the 2 or 3 variable equations using determinants. Students can think of the determinant as a magic box that helps us find the missing number in a puzzle. 
    
2. Parents should make sure your child understands what solving a system of equations means in simple terms. It is nothing but finding where two or three lines intersect. Don't directly try to push your kids in memorizing formulas.
    
3. Use some visuals like a table for determinants, which will make cross-multiplication easier. Use the arrow method to draw arrows diagonally to remember which numbers multiply for \(D, D_x, D_y\).
    
4. Try to solve problems with simple numbers like 1, 2, 2, -1 to avoid fractions initially. Parents must also encourage them to check the answers by substituting into the original equations. 
    

5. Make learning fun by playing games. Turn solving 2x2 systems into a “treasure hunt” where each determinant gives a clue. Keep some rewards like stickers or points for correct solutions.

We can apply Cramer’s rule when we need to solve multiple variables and equations. This mathematical method has several real-world applications in various fields, ranging from economics to engineering. 

• Economics: Economists use Cramer’s rule to find the equilibrium points where both supply and demand are equal. For example, to find out how demand and supply change in response to price can be figured using Cramer’s rule. 
   
• Engineering: Electrical engineers use Cramer's rule to solve multiple-loop and node equations and to find the unknown values in a complex circuit. For instance, they use this rule to solve for the current in three connected loops and three unknown currents. 
   
• Graphic design: designers use Cramer’s rule to change the positions of objects. They use it when they move, rotate, or resize 3D shapes. For example, if they know where an object ended up after being moved or turned, they can use Cramer’s rule to figure out where it started.
   
• Traffic or Distance Problems: when we have two cars traveling at different speeds, and they meet at a point, equations can represent distance = speed × time. We can solve for unknown speed or time using Cramer’s Rule.
   

• Mixing Solutions in Science: We can use Cramer's rule in some scenarios, like preparing lab mixtures or juice blends. For example: Mixing two solutions with different concentrations to get a desired concentration uses Cramer’s Rule to find how much of each solution is needed.