 Hi and welcome to the session. Today we will learn about properties of determinants. First property states that the value of the determinant remains unchanged if its rows, columns are interchanged. Symbolically we will write it as R i interchanges with C i for all i. From this property we get that for a square matrix A if we interchange all the rows with the columns then we will get its transpose. So that means determinant of A will be e equal to determinant of A transpose. The second property is if any two rows, columns of a determinant are interchanged then sign of determinant changes. That is if the sign of determinant is plus determinant it will change to minus and vice versa. Symbolically we can denote the interchange of rows as R i interchanges with R j and interchange of columns as C i interchanges with C j. Now let's see third property. This states that if any two rows or columns of a determinant identical that means all corresponding elements are same then value of determinant is 0. Now the fourth property is if each element of a row or a column of a determinant is multiplied by a constant K then its value gets multiplied by the constant K. From this property we get that we can take out any common factor any one row any one column of a given determinant. Also we get corresponding elements of any columns of a determinant are proportional that is in the same ratio then its value is 0. That is if we will take out the common factor of any one row or any one column outside the determinant then we will get either two identical rows or two identical columns then by property 3 we will get its determinant as 0. Now next property that is fifth property is if some or all elements of a row or column of a determinant are expressed then the determinant can be expressed or more determinants. Now the sixth property is each element of any row of a determinant the equal multiples corresponding elements other column are added then value of determinant remains the same that is if we change R i to R i plus K times R j or we change C i to C i plus K times C j then the value of determinant will remain same. Now let us take one example for this here we need to prove this determinant equal to 0. So let us start with the determinant itself the given determinant can be written like this here we have just simplified the column 3. Now we will change C 3 2 C 3 plus C 2 and thus the given determinant will be equal to now here C 1 will remain as it is C 2 will also be as it is now in C 3 we are adding C 2 so it will be A B plus B C plus C A this will be A B plus B C plus C A now here we will get A B plus B C plus C A here we have used property 6 as we have added to each element of column 3 the corresponding elements of column 2 multiplying them by 1. Now as we can see that all the elements of column 3 are same so we can take them common outside the determinant and this will be equal to A B plus B C plus C A into the determinant here again C 1 will remain as it is C 2 will also be same now as we have taken these elements common so we are left with 1 1 1 here we have used property 4. Now in this determinant C 1 and C 3 are identical so that means the value of this determinant will be equal to 0 so we get A B plus B C plus C A into 0 here we have used property 3 so this will be equal to 0 hence we have proved that the given determinant is equal to 0 with this we finished this session hope you must have understood all the properties goodbye take care and keep smiling.