 Hi and welcome to the session. Today we will learn about transpose of A matrix. If A given by aij bn m by n matrix then the matrix by interchanging the rows and columns of matrix A is called transpose of matrix A and the transpose of matrix A is denoted by, so symbolically if A is a matrix given by aij of order m by n then A transpose will be the matrix given by aji of order n by m. For example, for the given matrix A, A transpose will be 351-1. Now let's move on to the properties of transpose of the matrices. A matrices A and B of suitable order, the following properties are true. So first property is transpose of A transpose is equal to A itself. Second property is k into A whole transpose is equal to k into A transpose where k is any constant. A plus B whole transpose is equal to A transpose plus B transpose and the last that is for this A into B transpose is equal to B transpose into A transpose. Now our next topic is symmetric to symmetric matrices. So first of all let us see what is a symmetric matrix? A square matrix given by aij is said to be symmetric A transpose is equal to A. Symbolically we can say that the matrix A is symmetric if the matrix given by aij is equal to the matrix given by aji for all possible values of i and j. Now let's move on to skew symmetric matrix our square matrix A given by aij is said to be symmetric if A transpose is equal to minus A. Symbolically we have A is a skew symmetric matrix if the matrix given by aji is equal to the minus of the matrix given by aij for all possible values of i and j. Also all the diagonal elements of a skew symmetric matrix 0. Now let's see some results. So first result is for a square matrix with real number entries A plus A transpose is a symmetric matrix A minus A transpose is a skew symmetric matrix. The second result is a square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Let's take one example. Here we are given a matrix A and we need to express it as a sum of symmetric and skew symmetric matrix. Now from result one we have A plus A transpose is a symmetric matrix and A minus A transpose is a skew symmetric matrix. So we can express A as 1 by 2 into A plus A transpose plus 1 by 2 into A minus A transpose where this will be a symmetric matrix and this will be a skew symmetric matrix. So let us suppose this is equal to P and this is equal to Q. So this implies A can be expressed as P plus Q where P is a symmetric matrix and Q is a skew symmetric matrix. So for this first of all we need to find A transpose and this will be equal to 351 minus 1. Now let us find out A plus A transpose and this will be equal to 315 minus 1 plus 351 minus 1 which is equal to 666 minus 2. Now P equal to half of A plus A transpose will be equal to 333 minus 1. Now A minus A transpose will be equal to 315 minus 1 minus 351 minus 1 which is equal to 0 minus 440. So Q equal to half of A minus A transpose is equal to 0 minus 220. So let us express A as a sum of symmetric and skew symmetric matrix that is P plus Q. So this will be equal to 333 minus 1 plus 0 minus 220 where this is a skew symmetric matrix and this is a skew symmetric matrix. Now here this is a skew symmetric matrix and as we can notice that the diagonal elements of this skew symmetric matrix are all 0. With this we finish this session. Hope you must have understood all the concepts. Goodbye, take care and keep smiling.