 Hi and welcome to the session. I am Shashi. Let us do one question. Question is express the following matrices as a sum of a symmetric and a skew symmetric matrix. The given matrix is 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. First of all let us understand the key idea to solve the given question. Any square matrix A is said to be skew symmetric if a transpose is equal to minus a. That is the transpose of a matrix is equal to negative of itself. Then any square matrix A is said to be symmetric if a transpose is equal to a. That is the transpose of a matrix is equal to itself. Let us now start with the solution. Let A be equal to the given matrix. This is 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. Now we can find a transpose by interchanging the rows and columns of A. So we get a transpose is equal to 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. Now we will find a plus a transpose. You know a plus a transpose is equal to matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3 plus matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. Now we will add the corresponding elements of the two matrices to find a plus a transpose. So a plus a transpose is equal to 6 plus 6 is 12 minus 2 plus minus 2 is minus 4 2 plus 2 is 4 minus 2 plus minus 2 is minus 4 3 plus 3 is 6 minus 1 plus minus 1 is minus 2 2 plus 2 is 4 minus 1 plus minus 1 is minus 2 3 plus 3 is 6. So a plus a transpose is equal to matrix 12 minus 4 4 minus 4 6 minus 2 4 minus 2 6. Now let p is equal to half multiplied by a plus a transpose. So to find p we will multiply every element of a plus a transpose with half. So p is equal to half multiplied by the matrix 12 minus 4 4 minus 4 6 minus 2 4 minus 2 6. So we get p is equal to matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. You know half multiplied by 12 is 6, half multiplied by minus 4 is minus 2, half multiplied by 4 is 2, half multiplied by minus 4 is minus 2, half multiplied by 6 is 3, half multiplied by minus 2 is minus 1, half multiplied by 4 is 2, half multiplied by minus 2 is minus 1, half multiplied by 6 is equal to 3. So we get p is equal to matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. Now let us find out p transpose. p transpose is equal to 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. So p transpose is equal to matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. We can obtain p transpose by interchanging the rows and columns of p. Now we can see clearly that p and p transpose are exactly same. So we can write p is equal to p transpose. Therefore we get p is a symmetric matrix as we have already read in key idea if transpose of a matrix is equal to itself then it is a symmetric matrix. So p is a symmetric matrix. Now we know A transpose is equal to matrix 6 minus 2 2 minus 2 3 minus 1 2 minus 1 3. So minus A transpose is equal to matrix minus 6 2 minus 2 2 minus 3 1 minus 2 1 minus 3. To obtain minus A transpose we had multiplied every element of A transpose with minus 1. We know minus 1 into 6 is minus 6 minus 1 into minus 2 is 2 minus 1 into 2 is minus 2 minus 2 into minus 1 is 2 3 into minus 1 is minus 3 minus 1 into minus 1 is 1 2 multiplied by minus 1 is minus 2 minus 1 multiplied by minus 1 is 1 3 multiplied by minus 1 is minus 3. So minus A transpose is equal to this matrix. Now we know A minus A transpose is equal to A plus minus A transpose, right? So A minus A transpose is equal to matrix, A we know is the matrix 6 minus 2, 2 minus 2, 3 minus 1, 2 minus 1, 3 plus minus A transpose is the matrix, minus 6, 2, minus 2, 2, minus 3, 1, minus 2, 1, minus 3. Now we will add the corresponding elements of the two matrices. So we get A minus A transpose is equal to 6 plus minus 6 is 0, minus 2, plus 2 is 0, 2 plus minus 2 is 0, minus 2, plus 2 is 0, 3 plus minus 3 is 0, minus 1, plus 1 is 0, 2 plus minus 2 is 0, minus 1, plus 1 is 0, 3 plus minus 3 is 0. So A minus A transpose is equal to 0 matrix as all the elements of the matrix are 0. Now let Q is equal to half multiplied by A minus A transpose. So we can get Q by multiplying half with the matrix of A minus A transpose. A minus A transpose is a 0 matrix. So we get half multiplied by 0 matrix equal to 0 matrix. So Q is equal to 0 matrix. Now Q is equal to 0 matrix. So Q transpose will also be equal to 0 matrix only. We can write Q transpose is equal to minus Q multiplied by A minus A transpose is a skew symmetric matrix. Let us find out P plus Q. We know P is equal to matrix 6 minus 2, 2, minus 2, 3, minus 1, 2, minus 1, 3 and Q is a 0 matrix. P plus Q is equal to matrix P itself that is 6 minus 2, 2, minus 2, 3, minus 1, 2, minus 1, 3. But we can see P plus Q is exactly equal to matrix of A. So we can write P plus Q is equal to A. But P is a symmetric matrix and Q is a skew symmetric matrix. So we can represent A as a sum of symmetric and a skew symmetric matrix. So our final answer is A is equal to matrix 6 minus 2, 2, minus 2, 3, minus 1, 2, minus 1, 3 plus matrix 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 where the first matrix is the symmetric matrix and the second matrix is the skew symmetric matrix. So this completes the session. Hope you understood the session. Take care and goodbye.