 Hi and welcome to the session. Let us discuss the following question which is let A be the matrix of order 3 by 3 with the elements 3, 2, 5, 4, 1, 3, 0, 6, 7, express A as some of 2 matrices such that one is symmetric and the other is skew symmetric. Now before moving on to the solution let's recall what is a symmetric and skew symmetric matrix. Any square matrix A is said to be symmetric if A transpose is equal to A and any square matrix A is said to be skew symmetric if A transpose is equal to minus A. Now let us also recall an important theorem which will be very helpful for this question that is for any square matrix A with real number entries A plus A transpose is a symmetric matrix A minus A transpose is a skew symmetric matrix. This is the key idea for this question. Now let's see its solution. We are given a matrix A of order 3 by 3 with elements 3, 2, 5, 4, 1, 3, 0, 6, 7. So first of all we will find out A transpose. Now transpose of a matrix is obtained by interchanging the rows and columns of that matrix. So here A transpose will be a matrix of order 3 by 3 with elements 3, 4, 0, 2, 1, 6, 5, 3, 7. Now by the key idea we know that A plus A transpose is a symmetric matrix and A minus A transpose is a skew symmetric matrix and we can write the matrix A as 1 by 2 into A plus A transpose plus 1 by 2 into A minus A transpose because this will be equal to A itself. So here 1 by 2 into A plus A transpose is a symmetric matrix and 1 by 2 into A minus A transpose is a skew symmetric matrix. Thus we can express A as a sum of symmetric and skew symmetric matrix. So that means here we need to find two matrices that is the matrix P which is equal to 1 by 2 into A plus A transpose and matrix Q which is given by 1 by 2 into A minus A transpose. So first of all let us find out A plus A transpose which will be equal to the matrix A that is the matrix of order 3 by 3 with elements 3, 2, 5, 4, 1, 3, 0, 6, 7 plus the matrix A transpose with elements 3, 4, 0, 2, 1, 6, 5, 3, 7. Now we know that to add two matrices we are there corresponding elements. So this will be equal to the matrix of order 3 by 3 with elements 3 plus 3 that is 6, 2 plus 4 that is 6, 5 plus 0 that is 5, 4 plus 2, 6, 1 plus 1, 2, 3 plus 6, 9, 0 plus 5, 5, 6 plus 3, 9 and lastly 7 plus 7 that is 14. Now the matrix P is given by 1 by 2 into A plus A transpose. So the matrix P will be equal to 1 by 2 into A plus A transpose that is the matrix of order 3 by 3 with elements 6, 6, 5, 6, 2, 9, 5, 9, 14 and to multiply a matrix with a scalar we multiply each element of that matrix by the scalar that is 1 by 2. So here we will get the matrix of order 3 by 3 with elements 3, 3, 5 by 2, 3, 1, 9 by 2, 5 by 2, 9 by 2 and 7. Now let us check whether the matrix P is symmetric or not. So let us find out P transpose which will be a matrix of order 3 by 3 with elements 3, 3, 5 by 2, 3, 1, 9 by 2, 5 by 2, 9 by 2, 7 which is equal to the matrix P itself. Thus here the matrix P is a symmetric matrix. Now let us find out the matrix A minus A transpose. So here this will be equal to the matrix A that is the matrix of order 3 by 3 with elements 3, 2, 5, 4, 1, 3, 0, 6, 7 minus A transpose that is the matrix of order 3 by 3 with elements 3, 4, 0, 2, 1, 6, 5, 3, 7. Now here we will subtract each element of matrix A transpose from the corresponding elements of matrix A and thus this will be equal to the matrix of order 3 by 3 with elements 3 minus 3 that is 0, 2 minus 4, minus 2, 5 minus 0, 5, 4 minus 2, 2, 1 minus 1, 0, 3 minus 6, minus 3, 0 minus 5, minus 5, 6 minus 3, 3 and 7 minus 7, 0. Now the matrix Q is given by 1 by 2 into A minus A transpose. So let us find out the matrix Q which is equal to 1 by 2 into the matrix A minus A transpose which is this one. So Q will be equal to 1 by 2 into the matrix of order 3 by 3 with elements 0, minus 2, 5, 2, 0, minus 3, minus 5, 3, 0. And here also we will multiply each element of this matrix by the scalar 1 by 2. So this will be equal to the matrix of order 3 by 3 with elements 0, minus 1, 5 by 2, 1, 0, minus 3 by 2, minus 5 by 2, 3 by 2, 0. Now let us take whether Q is this Q symmetric matrix or not. So let us find Q transpose which will be given by the matrix of order 3 by 3 with elements 0, 1, minus 5 by 2, minus 1, 0, 3 by 2, 5 by 2, minus 3 by 2, 0 which is equal to minus Q thus Q is a Q symmetric matrix. Now let us find the matrix given by P plus Q. So this will be the matrix of order 3 by 3 with elements 3, 3, 5 by 2, 3, 1, 9 by 2, 5 by 2, 9 by 2 and 7 plus the matrix of order 3 by 3 with elements 0, minus 1, 5 by 2, 1, 0, minus 3 by 2, minus 5 by 2, 3 by 2 and 0. And this will be equal to the matrix of order 3 by 3 with elements 3 plus 0, 3, 3 plus minus 1, 2, 5 by 2 plus 5 by 2 that is 5, 3 plus 1, 4, 1 plus 0, 1, 9 by 2 plus minus 3 by 2 that is 3, 5 by 2 plus minus 5 by 2, 0, 9 by 2 plus 3 by 2 that is 6, 7 plus 0, 7 which is equal to the matrix A thus the matrix A can be expressed as the matrix of order 3 by 3 with elements 3, 3, 5 by 2, 3, 1, 9 by 2, 5 by 2, 9 by 2, 7 plus 3 by 3 with elements 0, minus 1, 5 by 2, 1, 0, minus 3 by 2, minus 5 by 2, 3 by 2, 0 where this is a symmetric matrix and this is a skew symmetric matrix thus we have expressed A as the sum of a symmetric and a skew symmetric matrix. With this we finish this question and the session as well. Hope you must have understood the question. Goodbye, take care and have a nice day.