Properties of Determinants And Some Important Determinants to Remember

Properties of Determinants:-

1. Determinant of a matrix is same as the determinant of its transpose.
2. If two rows or columns of a determinant are interchanged the determinant changes its sign.
3. If the elements of a row (column) of a determinant are multiplied by a constant K, then the determinant will be multiplied by the same constant. For example,

If A is a square matrix of order n, then

4. If all the elements of a row (column) of a determinant are zeros, then the value of that determinant is also zero.
5. If two rows (columns) of a determinant are equal then value of that determinant is zero.
Note:- If two rows (2 columns) of a determinant are proportionate then its value is zero.
6. If the elements of a row (column) of a determinant are sums of two elements then the determinant can be expressed as the sum of two determinants. That is for example,
7. If the elements of a row (column) of a determinant are added to or subtracted from the corresponding elements of some other row (column) then the determinant remains unchanged.
8. If the products of the elements of a row (or column) of a determinant with a constant K are added to the corresponding elements of some other row (or column), then the determinant remains unchanged.
9. Sum of the products of the elements of a row in a square matrix and the co-factors of the corresponding elements of some other row (column) is zero.
10. If the rows or columns of a determinant are changed without disturbing a cyclic order, then the determinant remains unchanged. That is,
11. Determinant of a null matrix is 1.
12. Determinant of a null matrix of the order 3X3 is zero.