Consider the following system of linear equations: 2x + 3y = 8 4x + 6y = 16 What is the determinant of the coefficient matrix, and what does it imply about the system's solution?

The coefficient matrix is: A = [2 3] [4 6] The determinant of a 2 × 2 matrix is calculated as: Det(A) = (2 × 6) − (3 × 4) = 12 − 12 = 0 Since: Det(A) = 0 the coefficient matrix is singular and therefore does not have an inverse. Now observe the two equations: 2x + 3y = 8 4x + 6y = 16 The second equation is exactly twice the first equation: 2 × (2x + 3y = 8) gives: 4x + 6y = 16 Therefore, both equations represent the same relationship. Hence, the equations are linearly dependent and the system has infinitely many solutions.