 Hi and welcome to the session. Today we will learn about operations on matrices. First of all we will learn addition of matrices. Of two matrices is a matrix obtained by adding the corresponding elements of the two matrices. Also the two matrices should be same order. For example suppose we are given two matrices a and b and we need to find their sum that is a plus b. So first of all let us take their order. Now the order of matrix a is 2 by 3 and the order of matrix b is also 2 by 3 so that means we can add them. Now a plus b will be equal to 2 plus 0 that is 2 then 6 plus minus 9 that is minus 3 5 plus 2 7 minus 3 plus 11 that is 8 then 8 plus 1 9 and lastly 1 plus 4 that is 5. So this is the required matrix. Now our next topic is multiplication of a matrix by a scalar. Suppose we are given a matrix a given by aij of order myn and we want to multiply the given matrix a by a scalar k then we will multiply the scalar k with each element aij of the matrix a and we will get a new matrix of order m by n. Let us take one example for this. Here we are given a matrix a and the scalar k equal to 2 and we need to multiply k with the matrix a. So k into a will be equal to 2 into 2 that is 4 then 2 into 6 12 then 10 minus 6 16 and 2. So this is equal to 2 into the matrix a. Now let us see what is negative of a matrix. Suppose we are given a matrix a then the negative of matrix a will be minus a and minus a is minus 1 into the matrix a that is we will multiply minus 1 with each element of the matrix a. For example suppose we are given matrix a then negative of a that is minus a will be equal to minus 1 into the matrix a that is 2 5 minus 3 0. So we will multiply minus 1 with each element of matrix a and this will be minus 2 minus 5 3 0. Now comes difference matrices given by aij and b given by bij are the two matrices of same order say m by n then the difference of the two matrices a minus b will be given by a third matrix d which is given by dij such that dij will be equal to aij minus vij for all i and j. Also the order of matrix d will be m by n for example here we are given two matrices a and b of order 3 by 2 and we need to find the difference of a and b so a minus v will be equal to a matrix d which will be minus 1 then 6 minus 4 minus 5 minus 2 3 minus 5 0 minus 7 and 9 minus 13 which is equal to 1 2 minus 7 minus 2 minus 7 and minus 4 and this will also be of order 3 by 2. Now let's see properties of matrix addition. The addition of matrices satisfy the following properties. First property is commutative law given by aij and b given by bij are the two matrices of same order say of order m by n then a plus b will be equal to b plus a. Let's take an example for this. Suppose we are given two matrices a and b of same order and we want to show that a plus b is equal to b plus a. So first of all let us find a plus b this will be equal to 8 0 4 minus 2 plus 2 minus 2 4 2 which will be equal to 10 minus 2 8 0. Now b plus a will be equal to 2 minus 2 4 2 plus 8 0 4 minus 2 and this will also be equal to 10 minus 2 8 0. From here we can conclude that a plus b is equal to b plus a. So we can say that the addition of matrices is commutative. Similarly, addition of matrices satisfies associative law for any three matrices a given by aij b given by bij c given by cij of the same order say of order m by n a plus b plus c is equal to a plus b plus c that means addition of matrices is associative. Now third is existence of additive identity is a matrix given by aij of order m by n is a zero matrix of order m by n then a plus o will be equal to o plus a which is equal to a itself. In other words o is the additive identity for matrix addition. Now the fourth one is existence of additive inverse is any matrix given by aij of order m by n then we have another matrix s a given by minus aij of order m by n such that a plus minus a is equal to minus a plus a which is equal to o which is a null matrix of order m by n then s a is the additive inverse of matrix a or we can also call it as negative of matrix a. Now our next topic is properties of scalar multiplication of a matrix given by aij b given by bij are the two matrices of same order say of m by n k and l are the two scalars then first we have k into a plus b will be equal to k into matrix a plus k into matrix b and second we have k plus l into matrix a is equal to k into matrix a plus l into matrix a so these two properties hold good for any matrix. With this we finish this session hope you must have enjoyed it goodbye take care and keep smiling.