Properties of Determinant of Matrix

Sure, here are some properties of the determinant of a matrix:

1. Determinant of a Square Matrix: The determinant is only defined for square matrices (i.e., matrices with the same number of rows and columns).

2. Determinant of a Unit Matrix: The determinant of a unit or identity matrix is 1.

3. Determinant of a Diagonal Matrix: The determinant of a diagonal matrix is the product of its diagonal elements.

4. Determinant of a Scalar Multiple: If we multiply all elements of a row (or column) by a scalar, the determinant also gets multiplied by the same scalar.

5. Interchanging Rows or Columns: If we interchange any two rows (or columns), the sign of the determinant changes.

6. Adding Rows or Columns: If we add (or subtract) the elements of one row to another, the determinant remains the same.

7. Determinant of a Transpose: The determinant of a matrix remains the same if its transpose is taken.

8. Determinant of a Product: The determinant of the product of two matrices is equal to the product of their determinants (det(AB) = det(A) * det(B)).

9. Determinant of an Inverse: The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix (if the matrix is invertible).

10. Zero Determinant: If the determinant of a matrix is zero, the matrix does not have an inverse and is known as a singular matrix.

These properties can be very useful when calculating determinants or when trying to simplify complex matrix expressions.