Solved problems using Cramer's Rule: Detailed linear algebra examples

📚 Quick Study Guide

• 🔢 Cramer's Rule is a method for solving systems of linear equations using determinants.
• 📝 For a system of $n$ equations in $n$ variables, Cramer's Rule can be applied if the determinant of the coefficient matrix is non-zero.
• 🧮 Given a system $Ax = b$, where $A$ is the coefficient matrix, $x$ is the variable vector, and $b$ is the constant vector:
• ➗ The solution for each variable $x_i$ is given by $x_i = \frac{det(A_i)}{det(A)}$, where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the vector $b$.
• 💡 Key Formula: $x_i = \frac{det(A_i)}{det(A)}$

Practice Quiz

1. Question 1: What is the primary condition for applying Cramer's Rule to a system of linear equations?

   • A) The number of equations must be less than the number of variables.
   • B) The determinant of the coefficient matrix must be non-zero.
   • C) The system must have infinitely many solutions.
   • D) The equations must be non-linear.

2. Question 2: In Cramer's Rule, what does $A_i$ represent?

   • A) The original coefficient matrix.
   • B) The matrix formed by replacing the $i$-th row of $A$ with the constant vector $b$.
   • C) The matrix formed by replacing the $i$-th column of $A$ with the constant vector $b$.
   • D) The inverse of the coefficient matrix.

3. Question 3: Consider the system: $2x + y = 7$, $x - y = 2$. What is the determinant of the coefficient matrix $A$?

   • A) -3
   • B) 1
   • C) 3
   • D) 0

4. Question 4: Using the same system as above, what is the determinant of $A_x$ (replacing the x-column with the constants)?

   • A) 9
   • B) 14
   • C) 12
   • D) 16

5. Question 5: What is the solution for $x$ in the system from questions 3 & 4?

   • A) $x = 2$
   • B) $x = 3$
   • C) $x = 1$
   • D) $x = 4$

6. Question 6: For what type of systems is Cramer's rule most effective?

   • A) Very large systems of equations.
   • B) Systems with non-integer coefficients.
   • C) Small systems of linear equations with unique solutions.
   • D) Systems with infinitely many solutions.

7. Question 7: If det(A) = 0, what does this imply about the system of equations?

   • A) The system has a unique solution.
   • B) Cramer's Rule can still be applied.
   • C) The system has either no solution or infinitely many solutions.
   • D) The system is inconsistent.