Determinants of 2x2 Matrices: Can a System Have Multiple Solutions?

📌 CONCEPT: A system of linear equations represented by a 2x2 matrix A with determinant det(A) = 0 has either no solution or infinitely many solutions.

📐 RULE / FORMULA: The determinant of a 2x2 matrix A with integer entries is given by det(A) = ad - bc, where A = [[a, b], [c, d]]. A system of linear equations represented by A has multiple solutions if and only if det(A) = 0.

💡 WORKED EXAMPLE: Suppose we have two matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]] with integer entries. The determinant of A is det(A) = (1)(4) - (2)(3) = -2 and the determinant of B is det(B) = (5)(8) - (6)(7) = -22. Since det(A) = det(B) = 0, the system of linear equations represented by A and B has infinitely many solutions, but A ≠ B.

⚠️ COMMON MISTAKE: Students often confuse the condition det(A) = 0 with the system having a unique solution, but this is incorrect because a system of linear equations represented by A can have either no solution or infinitely many solutions when det(A) = 0.