Properties of Determinants:-
- Determinant of a matrix is same as the determinant of its transpose.
- If two rows or columns of a determinant are interchanged the determinant changes its sign.
- 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
- If all the elements of a row (column) of a determinant are zeros, then the value of that determinant is also zero.
- 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. - 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,
- 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.
- 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.
- 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.
- If the rows or columns of a determinant are changed without disturbing a cyclic order, then the determinant remains unchanged. That is,
- Determinant of a null matrix is 1.
- Determinant of a null matrix of the order 3X3 is zero.
Some Important Matrices Determinants to be Remembered for competitive exams:-
