The determinant of matrix [[a,b],[c,d]] is ad - bc. If ad - bc = 0, what does that say about the matrix?

• A. It is invertible
• B. It has full rank
• C. It has determinant greater than zero
• D. It is singular (non-invertible) or its rows are linearly dependent

Answer: (D)

When a determinant is zero, the linear transformation represented by the matrix cannot be inverted. For a 2x2 matrix, ad − bc = 0 means the two rows (and also the two columns) are linearly dependent, so one row is a multiple of the other. That dependency collapses area to zero and lowers the rank below the full size, making the matrix singular. In short, it is non-invertible. The determinant being zero rules out invertibility and identifies the dependence between rows or columns.