 Hi and welcome to the session. Today we will learn about elementary operation or transformation of a matrix. There are 6 operations or transformations on a matrix in which 3 are due to rows and 3 are due to columns and these are known as elementary operations or transformations. So let us see what are these 6 elementary operations or transformations. First one is the interchange of any 2 rows or columns. So suppose we want to interchange ith row with jth row then we will write ri interchanges with rj. Similarly if we want to interchange ith column with jth column then we will write it symbolically as ci interchanges with cj. For example if in this matrix we want to interchange r1 with r2 then we will get the matrix A as 1 3 2 5. Now the second elementary operation or transformation is the multiplication of the elements of any row or column by a non-zero number. Symbolically the multiplication of each element of ith row by a non-zero number k will be given as ri changes to k into ri. Similarly the multiplication of a non-zero number k with ith column will be given as ci changes to k into ci. For example if in this matrix we change c2 to 2 into c2 then we will get A as 2 1 column 1 will be as it is there will be changes in c2 so this will be 10 6. And the third transformation is the addition to the elements of any row the corresponding elements of any other row or column multiplied by any non-zero number that is if we add to the elements of ith row the corresponding elements of jth row multiplied by a non-zero number k then we will write ri changes to ri plus k into rj. Similarly if we add to the elements of ith column the corresponding elements of jth column when multiplied by a non-zero number k then we will get ci changes to ci plus k into cj. For example if in this matrix we want to change r1 to r1 plus 2 times r2 then we will get A as here 2 will change to 2 plus 2 times of 1 that is 4 and 5 will change to 5 plus 2 times of 3 that is 11 and r2 will remain as it is 1 3. Now let's move on to our next topic invertible matrices. Is it square matrix of order m if there exists another square matrix B of the same order m such that A into B is equal to B into A is equal to i that is identity matrix then B is called the inverse of matrix A is denoted by inverse in this case it is said to be invertible this is true for square matrices same order. Now there is one more point to be remembered that is B is inverse of A then A is also the inverse of B. Now let us quickly go through some properties. So first property states that the inverse of a square matrix exists is unique and the second property and B invertible matrices of the same order then A into B inverse is equal to B inverse into A inverse. Now the last topic is to find out inverse of a matrix A elementary operations we are given a matrix A and we want to find out A inverse using elementary row or column operations. Now if we want to find out A inverse using elementary row operations then we will write A equal to i into A where i is an identity matrix then we will apply a sequence of row operations on A equal to i into A till we get i equal to B into A the matrix B will be the inverse of A. Similarly if we want to find out A inverse using column operations then we will write A equal to A into i where i is the identity matrix then we will apply a sequence of column operations on A equal to A into i till we get i equal to A into B where B is the inverse of A. Now if in case after applying one or more elementary row or column operations on A equal to i into A or A equal to A into i we obtain all zeroes in one or more rows of the matrix A on left hand side then that means A inverse does not exist. Now let's take one example for this here we are given a matrix A and we need to find A inverse using elementary row operations. So we will write it in the form A equal to i into A so we will get the matrix A equal to identity matrix that is one zero zero one into matrix A. Now we will apply elementary row operations on this so as to get identity matrix on left hand side. So first of all we will try to change this element to one so let us take r1 changes to one upon two into r1 and thus we will get one by two one three equal to one by two zero zero one into A. Now we need to change this element to zero so we will apply the row operation r2 changes to r2 minus r1 and thus we will get one five by two zero one by two equal to one by two zero minus one by two one into A. Now we want to change this to one so let us apply the row operation r2 changes to two times r2 and this will give us one five by two zero one equal to one by two zero minus one two into A. Now lastly we need to change five by two to zero so for this we will change r1 to r1 minus five by two into r2 and thus we get one zero zero one equal to three minus five minus one two into A. So on left hand side we got the identity matrix so we can say that this is in the form i equal to B into A where B is the matrix three minus five minus one two and thus A inverse is given by the matrix B or we can say that A inverse is the matrix three minus five minus one two. So in same way we can find out A inverse using an inventory column operations. With this we finish this session hope you must have understood all the concepts goodbye take care and have a nice day.