 Hello and welcome to the session. Let's discuss the following question. It says, express the matrix A as the sum of symmetric and skew symmetric matrix. So let's now move on to the solution. The given matrix A is 2, 3, 2, minus 1, 1, minus 1, 0, 3, 0. Now we have to express this matrix as the sum of symmetric and skew symmetric matrix. Right? So the symmetric matrix of is given by A plus A transpose by 2. Now for this we first find A transpose and to find transpose of a matrix we just change rows by columns and columns by rows. So the first row becomes the first column of A transpose. Second row becomes the second column and the third row becomes the third column. So this is A transpose. Now we find A plus A transpose by 2 that is half of A plus A transpose. Now A is minus 1, 0, 3, 1, 3, 2, minus 1, 0 plus A transpose A transpose is 2, 3, 2, minus 1, 1, minus 1, 0, 3, 0. So this is equal to 1 by 2 of 2 plus 2 is 4, minus 1 plus 3 is 2, 0 plus 2 is 2, 2, 1 plus 1, 2, 3 plus minus 1 is 2, 2 plus 0 is 2, minus 1 plus 3 is 2, 0 plus 0 is 0. So this is equal to dividing each entry by 2 we have 2, 1, 1, 1, 1, 1, 1, 1, 1, 0. So this is the symmetric matrix of A. Symmetric matrix is the matrix where the transpose of the matrix is equal to the matrix itself. So this is a symmetric matrix. Now the skew symmetric matrix of A is given by half of A minus A transpose. So this is half of A minus A transpose. Now A is 2 minus 1, 0, 3, 1, 3, 2, minus 1, 0, minus A transpose A transpose is this that is 2, 3, 2, minus 1, 1, minus 1, 0, 3, 0. So now this is equal to half of 2 minus 2 is 0, minus 1 minus 3 is minus 4, 0 minus 2 is minus 2, 3 minus minus 1 is 4, 1 minus 1 is 0, 3 minus minus 1 is 4, 2 minus 0 is 2, minus 1 minus 3 is minus 4, 0 minus 0 is 0. So this is equal to dividing each entry by 2 we have 0 minus 2, minus 1, 2, 0, 2, 1, minus 2, 0. Now this is the skew symmetric matrix of A. Skew symmetric matrix is the matrix where the transpose of the matrix is equal to the negative of that matrix. So here transpose of this matrix will be the negative of this matrix. So now A which is the matrix 2 minus 1, 0, 3, 1, 2, minus 1, 0 this can be written as the sum of symmetric matrix which is A plus A transpose by 2 that is 2, 1, 1, 1, 1, 1, 1, 1, 0 plus A minus A transpose by 2 that is this matrix 0, minus 2, minus 1, 2, 0, 2, 1, minus 2, 0. So this is how we have expressed the matrix A as a sum of symmetric and skew symmetric matrix. So this completes the question and the session bye for now. Take care, have a good day.