Determinant of a 3x3 Matrix - Application?

📌 CONCEPT: A determinant of a 3x3 matrix can be used to determine the nature of the solution of a system of three linear equations in two variables, i.e., whether it has a unique solution, no solution, or infinitely many solutions.

📐 RULE / FORMULA: The determinant of the coefficient matrix of a system of three linear equations in two variables can be used to determine the nature of the solution. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system has either infinitely many solutions or no solution.

💡 WORKED EXAMPLE: Let's consider the following system of linear equations in two variables:

x + 2y = 4

3x + 6y = 12

The coefficient matrix is

egin{bmatrix} 1 & 2 \ 3 & 6 \\\end{bmatrix}. The determinant of the coefficient matrix is (1 * 6) - (2 * 3) = -6, which is non-zero. Therefore, the system has a unique solution.

⚠️ COMMON MISTAKE: Students often forget to check the determinant of the coefficient matrix or assume that a zero determinant always implies infinitely many solutions. They should remember that a zero determinant may also imply no solution, depending on the equations.