Prove that the diagonal elements of a skew-symmetric matrix are all zero

Prove that a matrix which is symmetric as well as skew-symmetric is a null matrix.

Express the matrix A as the sum of a symmetric matrix and a skew-symmetric matrix, where A=[(2,4,-6),(7,3,5),(1,-2,4)]

Answer the following questions: Let A be a skew-symmetric matrix of odd order. Write the value of absA .

What is the sum of the diagonal elements of the following square matrix? [(1, 0, 1),(0, 1,2),(3, 1,-2)] .

Prove tha tany square matrix can be expressed uniquely as the sum of a symmetric and a skew-symmetric matrix.

Prove that the perimeter of any quadrilateral is greater than the sum of the diagonals.

If A=[(1,5),(6,7)] , verify that A+A' is a symmetric matrix and A-A' is a skew-symmetric matrix.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squarea of its diagonals.

Let R be the relation defined in the set A={1,2,3,4,5,6,7,8,9} by R={(a,b,):" both a and b are either odd or even"} . Show that R is an equivalence relation. Further show that all the lements of the subset {1,3,5,7,9} are related to one another and all the lement of the subset {2,4,6,8} are related to one another but no element of the subset {1,3,5,7,9} is related to any element of the subset {2,4,6,8} .

Express [(2,3,4),(5,6,-2),(1,4,5)] as the sum of a symmetric matrix and a skew-symmetric matrix.