Properties of Determinants:

Let A is a determinant.

1. The value of a determinant does not change when rows and columns are interchanged

.i.e |A^T | = |A|

2. If any row( or columns) of determinant is completely zero then |A| = 0

3. If any two rows( columns) of a determinant are identical then |A| = 0

4. If any two rows or two columns of a determinant are interchanged then the value of determinant is multiplied by -1

5. If all elements of the one row (or one column) of a determinant are multiplied by same number k, then the value of the determinant is k times the value og given

    determinant.

6. If A is n rowed square matrix, and k is any scalar, then |kA| = kⁿ *|A|

7. |Aⁿ | = (|A|)ⁿ

8. |A^n-1 | = 1/|A|

9. Using the fact that A*adj(A) = |A|*I where I is an identity matrix

a. |adj A| = |A|^n-1 

b. |adj(adj A)| = |A|^(n-1)*(n-1)