Cramer's Rule for 3×3 Linear Systems

1️⃣ Understand when Cramer's Rule applies to larger systems

2️⃣ Learn the advantages of algorithmic approaches over substitution methods

3️⃣ Master 3×3 coefficient matrix extraction and determinant calculation

4️⃣ Apply column replacement method for 3×3 systems

5️⃣ Solve complete 3×3 linear systems systematically

① Cramer's Rule for Larger Systems

Cramer's Rule is a method for solving linear systems where the number of equations equals the number of variables. This means the coefficient matrix will be square: 2×2, 3×3, 4×4, and so on.

For Cramer's Rule: Number of Equations = Number of Variables

Cramer's Rule is particularly valuable in computer systems because its algorithms are clear and systematic. We can program the solution method into clear computer algorithms or codes, allowing the computer to proceed step by step until it reaches the final solution.

Advantage over Traditional Methods: The traditional substitution method becomes increasingly complex and error-prone as the number of equations grows. With more equations, we need more substitution steps, increasing the probability of computational errors.

② Setting Up the 3×3 System

Let's solve a 3×3 system using Cramer's Rule. We have three equations with three variables, so the coefficient matrix will be 3×3:

3×3 System Example

4x + 5y - 6z = -14

3x - 2y + 7z = 47

7x - 6y - 8z = 15

Notice that we have three equations and the number of equations equals the number of variables. This means we can solve this system using Cramer's Rule.

③ Extracting the Coefficient Matrix

From our system, we extract the coefficient matrix and calculate its determinant using the 3×3 determinant method we learned in previous lessons.

Coefficient Matrix =

\begin{pmatrix} 4 & 5 & -6 \\ 3 & -2 & 7 \\ 7 & -6 & -8 \end{pmatrix}

Calculating the Main Determinant

Step 1: Set up the 3×3 determinant

D = \begin{vmatrix} 4 & 5 & -6 \\ 3 & -2 & 7 \\ 7 & -6 & -8 \end{vmatrix}

Step 2: Apply cofactor expansion or the rule of Sarrus
Using the method from our previous 3×3 determinant lesson

Step 3: Calculate the result

D = 621

Main Determinant: D = 621

④ Applying Cramer's Rule for 3×3

Now we can set up the Cramer's Rule formulas for our three variables. Each variable equals a specific determinant divided by the main determinant:

3×3 Cramer's Rule Formulas

x = \frac{D_x}{D} = \frac{D_x}{621}

y = \frac{D_y}{D} = \frac{D_y}{621}

z = \frac{D_z}{D} = \frac{D_z}{621}

Now we need to determine what matrices to use for $D_x$ , $D_y$ , and $D_z$ .

⑤ Finding D_x (Determinant for x)

For $D_x$ , we replace the entire x-column with the constants from the right side of our equations.

Step-by-Step: Finding D_x

Original coefficient matrix:

\begin{pmatrix} 4 & 5 & -6 \\ 3 & -2 & 7 \\ 7 & -6 & -8 \end{pmatrix}

Replace the x-column with constants (-14, 47, 15):

D_x = \begin{vmatrix} -14 & 5 & -6 \\ 47 & -2 & 7 \\ 15 & -6 & -8 \end{vmatrix}

Calculate the 3×3 determinant:

D_x = 3105

Therefore:

x = \frac{D_x}{D} = \frac{3105}{621} = 5

⑥ Finding D_y (Determinant for y)

For $D_y$ , we replace the y-column with the constants from the right side.

Step-by-Step: Finding D_y

Replace the y-column with constants (-14, 47, 15):

D_y = \begin{vmatrix} 4 & -14 & -6 \\ 3 & 47 & 7 \\ 7 & 15 & -8 \end{vmatrix}

Calculate the 3×3 determinant:

D_y = -1242

Therefore:

y = \frac{D_y}{D} = \frac{-1242}{621} = -2

⑦ Finding D_z (Determinant for z)

Finally, for $D_z$ , we replace the z-column with the constants.

Step-by-Step: Finding D_z

Replace the z-column with constants (-14, 47, 15):

D_z = \begin{vmatrix} 4 & 5 & -14 \\ 3 & -2 & 47 \\ 7 & -6 & 15 \end{vmatrix}

Calculate the 3×3 determinant:

D_z = 2484

Therefore:

z = \frac{D_z}{D} = \frac{2484}{621} = 4

⑧ Final Solution and Verification

Let's summarize our complete solution and verify it works in all three original equations.

Complete Solution

x = 5, \quad y = -2, \quad z = 4

Verification:

Equation 1:

4(5) + 5(-2) - 6(4) = 20 - 10 - 24 = -14

✅

Equation 2:

3(5) - 2(-2) + 7(4) = 15 + 4 + 28 = 47

✅

Equation 3:

7(5) - 6(-2) - 8(4) = 35 + 12 - 32 = 15

✅

Summary of 3×3 Cramer's Rule Process:

Verify that number of equations = number of variables
Extract the 3×3 coefficient matrix
Calculate main determinant D using 3×3 methods
For each variable, replace its column with constants and calculate new determinant
Divide each variable's determinant by main determinant
Verify solution in all original equations

Summary of Key Points

1️⃣ Cramer's Rule works when #equations = #variables (square coefficient matrix)

2️⃣ It provides clear algorithms ideal for computer implementation

3️⃣ For 3×3 systems: extract coefficient matrix and calculate D using 3×3 methods

4️⃣ Replace columns systematically: x-column for D_x, y-column for D_y, z-column for D_z

5️⃣ Each variable = (its determinant) ÷ (main determinant)