 Hi, and welcome to the session. I'm Shashi. Let us do one question. Question is express the following matrices as the sum of symmetric and skew symmetric matrix. First part is matrix 3, 5, 1, minus 1. 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. Any square matrix A is said to be symmetric if A transpose is equal to A. So, we should remember that symmetric matrix should fulfill the condition A transpose must be equal to A. That is, transpose of A must be equal to itself. And a skew symmetric matrix must fulfill the condition that A transpose is equal to negative of A. Right? This is the key idea to solve the given question. Now, let us start with the solution. Let A is equal to matrix given in the question that is 3, 5, 1, minus 1. So, A transpose is equal to 3, 1, 5, minus 1. A transpose can be obtained by interchanging the rows and columns of A. Now, we will find A plus A transpose. So, A plus A transpose is equal to matrix 3, 5, 1, minus 1, plus 3, 1, 5, minus 1 matrix. Now, to find A plus A transpose, we will add the corresponding elements of two matrices. So, we get 3 plus 3 is 6, 5 plus 1 is 6, 1 plus 5 is 6, minus 1 plus minus 1 is minus 2. Now, let P is equal to half multiplied by A plus A transpose. So, we get P is equal to half multiplied by 6, 6, 6, minus 2 matrix. Now, this is further equal to matrix 3, 3, 3, minus 1. We will multiply half with every element of the matrix. So, we get the matrix 3, 3, 3, minus 1. Now, let us find out P transpose. P transpose can be obtained by interchanging the rows and columns of the matrix P. So, it would be 3, 3, 3, minus 1, which is exactly same as the matrix P. So, we get P is equal to P transpose. Now, since P is equal to P transpose, we can write P is symmetric matrix as we had already read in the key idea. When transpose of matrix is equal to itself, then it is called symmetric matrix. So, P is a symmetric matrix. We know A transpose is equal to 3, 1, 5, minus 1 matrix. So, minus A transpose would be equal to minus 3, minus 1, minus 5, 1. To obtain minus A transpose, we had multiplied every element of A transpose with minus 1. So, we get minus A transpose equal to matrix minus 3, minus 1, minus 5, 1. Now, we know A minus A transpose is equal to A plus minus A transpose. So, we can write A minus A transpose is equal to A is the matrix 3, 5, 1, minus 1. And minus A transpose is the matrix minus 3, minus 1, minus 5, 1. Now, we will add the corresponding elements of the two matrices and get A minus A transpose equal to 3 plus minus 3 is 0, 5 plus minus 1 is 4, 1 plus minus 5 is minus 4, minus 1 plus 1 is 0. Now, let Q is equal to half multiplied by A minus A transpose. So, we get Q is equal to half multiplied by matrix 0, 4, minus 4, 0. Now, we will multiply every element of the matrix with half. So, we get Q is equal to half multiplied by 0 is 0, half multiplied by 4 is 2, half multiplied by minus 4 is minus 2 and half multiplied by 0 is 0. So, Q is equal to matrix 0, 2, minus 2, 0. Now, we can find Q transpose by interchanging rows and columns of Q. So, Q transpose is equal to 0 minus 2, 2, 0. Now, we can write Q transpose equal to minus 1 multiplied by matrix 0, 2, minus 2, 0. We had taken minus 1 as a common factor from every element of the matrix Q transpose. So, we can see this matrix is same as the matrix Q. So, we can replace it by Q. So, we get Q transpose is equal to minus 1 multiplied by Q or Q transpose is equal to minus Q. Now, since Q transpose is equal to minus Q, Q is a skew symmetric matrix. We know matrix is skew symmetric if the negative of the matrix is equal to its transpose. Now, let us find P plus Q. P plus Q is equal to matrix 3, 3, 3 minus 1 plus 0, 2, minus 2, 0. Now, we will add the corresponding elements of the two matrices. So, we get 3 plus 0 is equal to 3, 3 plus 2 is equal to 5, 3 plus minus 2 is equal to 1, minus 1 plus 0 is minus 1. So, P plus Q is equal to 3, 5, 1 minus 1. But we can see P plus Q is exactly same as the matrix A. So, we get P plus Q equal to A. So, we can write it equal to A. So, therefore, P plus Q is equal to A. We know P is a symmetric matrix and Q is a skew symmetric matrix. So, A is equal to the sum of symmetric matrix P and the skew symmetric matrix Q. So, this is our required answer. Hope you understood the session. Take care and goodbye.