 Hello and welcome to the session. The given question says, express the matrix A as the sum of symmetric and a skew symmetric matrix. So first let us learn that for a square matrix A with real entries, A plus A transpose is the symmetric matrix and A minus A transpose is the skew symmetric matrix and the square matrix A can be expressed as the sum of skew symmetric matrix and the symmetric matrix. That is, A can be expressed as A plus A transpose divided by 2 plus A minus A transpose divided by 2. So let us start with the solution and we are given the square matrix A with elements 1, 3, 5 in the first row minus 6, 8, 3 in the second row minus 4, 6, 5 in the third row. First let us find A transpose which we get on interchanging the rows and columns. Then it will be a 3 cross 3 matrix and 1 minus 6 minus 4 will be the elements of the first row. Then we have 3, 8, 6 and 5, 3, 5. First let us find the symmetric matrix and it is given by A plus A transpose and now let us find A plus A transpose. So on adding these 2 we get 2, minus 3, 1, minus 3, 16, 9, 1, 9 and 10. So let us check whether this is a symmetric matrix or not, let us name this matrix as B. Now B will be a symmetric matrix if we have B is equal to B transpose. So let us find B transpose, its element will be 2 minus 3, 1 in the first row, minus 3, 16 and 9 in the second row and in the third row we have 1, 9 and 10 and I am comparing this with the matrix B which is A plus A transpose, we see that B transpose is equal to B. So this implies that A plus A transpose is a symmetric matrix. Now let us find A minus A transpose, now 1 minus 1 gives 0, 3 minus of minus 6 gives 9, similarly here also we have 9 and we have minus 9, 0, minus 3, minus 9, 3 and 0. Let us name this matrix as C, now let us check whether A minus A transpose is a skew symmetric matrix or not. So for a matrix to be skew symmetric we must have C equal to minus of C transpose. So let us find first what is C transpose, here C transpose is 0, minus 9, minus 9, then we have 9, 0, 3, then we have 9, minus 3, 0 and this is equal to minus taking common we will get the matrix C. So here we have C is equal to minus C transpose and this implies that A minus A transpose is a skew symmetric matrix. Now we will show that it can be expressed as A plus A transpose divided by 2 plus A minus A transpose divided by 2. So starting with the right hand side we have A plus A transpose divided by 2 plus A minus A transpose divided by 2 where A plus A transpose is a symmetric matrix and A minus A transpose is a skew symmetric matrix. So A plus A transpose we have this matrix so dividing each element by 2 we have 1, minus 3 by 2, 1 by 2, minus 3 by 2, 8, 9 by 2, 1 by 2, 9 by 2 and 5 plus A minus A transpose divided by 2. So here also dividing each element by 2 we have 0, 9 by 2, 9 by 2, minus 9 by 2, 0 minus 3 by 2, minus 9 by 2, 3, by 2 and 0. This is further equal to, when adding both the matrices 1 plus 0 gives 1, minus 3 by 2, plus 9 by 2 gives 6 by 2 which is equal to 3, then half plus 9 by 2 gives 10 by 2 which is equal to 5 and we have minus 6 and we have 8, here adding 9 by 2 with minus 3 by 2 we get 6 by 2 which is equal to 3, here we have minus 4, here we have 6 and 5. Now on comparing this with the given matrix A we find that this is equal to the matrix A and thus A is equal to A plus A transpose divided by 2 plus A minus A transpose divided by 2 and thus this can be expressed as the sum of a symmetric matrix and a spew symmetric matrix. So this completes the session, bye and take care.