 Hello and welcome to the session. The given question says express the following metrics as the sum of symmetric and skew symmetric metrics. So first let us learn that for any square matrix A with real entries A plus A transpose is a symmetric and A minus A transpose is a skew symmetric matrix that is A can be written as 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 let's start with the solution and let us denote the given matrix by A. So A is equal to 1, 3, 5, minus 6, 8, 3, minus 4, 6, 5. First let us find A transpose. Now this can be written as 1, minus 6, minus 4, 3, 8, 6, 5, 3, 5. Now we shall try to write A can be written as A plus A dash divided by 2 and Q is equal to A minus A transpose divided by 2. Sorry this is also A transpose. So we have to write A in the form of P plus Q to show that A can be written as the sum of symmetric and a skew symmetric matrix. So first let us find P A plus A transpose. So A is 1, 3, 5, minus 6, 8, 3, minus 4, 6, 5 plus A transpose is 1, minus 6, minus 4, 3, 8, 6 and 5, 3, 5. This is further equal to half. On adding we have 2. On adding 3 with minus 6 we have minus 3, 5 plus or minus 4 is 1, minus 6, plus 3 is minus 3, 8 plus 8 is 16, 3 plus 6 is 9, minus 4 plus 5 is 1, 6 plus 3 is 9 and 5 plus 5 is 10. So this is further equal to 2 divided by 2 is 1, minus 3 by 2, 1 by 2, then minus 3 by 2, 16 by 2 gives 8 and 9 by 2, A 1 by 2, 9 by 2 and 10 by 2 is 5. So this is the value of P. Now let us find Q. So Q is half of A minus A transpose. So A is the matrix given to us which is 1, 3, 5, minus 6, 8, 3, minus 4, 6, 5, minus A transpose and A transpose is 1, minus 6, minus 4, 3, 8, 6 and 5, 3, 5. Let us solve it. So this will be equal to half. Now 1 minus 1 is 0, 3 minus of minus 6 is 3 plus 6 which gives 9, 5 plus 4 gives 9, minus 6 minus 3 gives minus 9, 8 minus 8 gives 0, 3 minus 6 gives minus 3, minus 4 minus 5 gives minus 9, 6 minus 3 gives 3 and 5 minus 5 gives 0. So this is further equal to 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. So this is the matrix Q. Now we will show that P is a symmetric matrix. That is we will show that P transpose is equal to P and Q is a skew symmetric matrix. That is we will show that Q transpose is equal to minus Q. Let us start with P now. First let us find P transpose. Now P transpose is equal to, this is the matrix P. So P transpose is 1 minus 3 divided by 2, half minus 3 divided by 2, 8, 9 by 2, half, 9 by 2, 5. Now on comparing it with P we find that P transpose is equal to P which implies that P is a symmetric matrix. Now let us show that Q is a skew symmetric matrix. Now Q transpose is equal to 0, minus 9 by 2, minus 9 by 2, 9 by 2, 0, 3 by 2, 9 by 2, minus 3 by 2, 0. Taking minus sign common we have minus 0, 9 by 2, 9 by 2, minus 9 by 2, 0, minus 3 by 2, minus 9 by 2, plus 3 by 2 and 0. So this is equal to minus of Q which further implies that Q is a skew symmetric matrix. Now let us find P plus Q. Now matrix P is 1 minus 3 by 2, half minus 3 by 2, 8, 9 by 2, half, 9 by 2, 5 and Q matrix is 0, 9 by 2, 9 by 2, minus 9 by 2, 0, minus 3 by 2, minus 9 by 2, 3 by 2, 0. Now adding the two matrices we have 1 plus 0 is 1, minus 3 by 2, plus 9 by 2 is 6 by 2 which is equal to 3, half plus 9 by 2 gives 10 by 2 which is equal to 5, minus 3 by 2 and minus 9 by 2 gives minus 12 by 2 which is equal to minus 6 and similarly we have 8, 3, 4 with a negative sign, 6 and 5. And this is matrix A. So this implies that A is equal to P plus Q well, P is a symmetric matrix, Q is a skew symmetric matrix. So this completes the session, buy and take care.