Which statement about the determinant is true?

Which statement about the determinant is true?

• A. It can be positive, negative, or zero
• B. It can only be positive
• C. It is always zero
• D. It is always positive

Answer: (A)

The determinant measures how a linear transformation scales area (and, in higher dimensions, volume) and whether it preserves or reverses orientation. Because of that, its value can be positive, negative, or zero. For example, the identity matrix scales area by 1 and preserves orientation, giving a positive determinant. A matrix that flips orientation, like a reflection, has a negative determinant. If the rows (or columns) are proportional, the transformation collapses area to zero, so the determinant is zero. A simple zero example is [ [1, 2], [2, 4] ], where the second row is a multiple of the first. Therefore, the determinant is not restricted to one sign and can take positive, negative, or zero values depending on the matrix.

The determinant measures how a linear transformation scales area (and, in higher dimensions, volume) and whether it preserves or reverses orientation. Because of that, its value can be positive, negative, or zero. For example, the identity matrix scales area by 1 and preserves orientation, giving a positive determinant. A matrix that flips orientation, like a reflection, has a negative determinant. If the rows (or columns) are proportional, the transformation collapses area to zero, so the determinant is zero. A simple zero example is [ [1, 2], [2, 4] ], where the second row is a multiple of the first. Therefore, the determinant is not restricted to one sign and can take positive, negative, or zero values depending on the matrix.