Determinants are mathematical quantities associated with square matrices. They have several important properties that are useful in various mathematical applications.

1. Determinant of a Scalar Multiple: If a matrix A is multiplied by a scalar k, the determinant of the resulting matrix is k times the determinant of A. For example, if A = [[2, 3], [4, 6]], then the determinant of A is (2 * 6) - (3 * 4) = 0, and if we multiply A by 2, we get [[4, 6], [8, 12]], which has a determinant of (4 * 12) - (6 * 8) = 0.

2. Determinant of a Transpose: The determinant of a matrix A is equal to the determinant of its transpose, denoted as det(A) = det(A^T). This property can be useful in certain calculations.

3. Determinant of a Product: The determinant of a product of two matrices is equal to the product of their determinants, i.e., det(AB) = det(A) * det(B). This property allows us to simplify calculations involving determinants of matrix products.

4. Determinant of an Inverse: If A is an invertible matrix, then the determinant of its inverse is equal to the reciprocal of the determinant of A, i.e., det(A^-1) = 1 / det(A). This property is crucial in solving systems of linear equations using matrix inversion.

These properties provide valuable insights and shortcuts in solving various mathematical problems involving determinants.