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 cofactors 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.
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