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n×n-matrix M is symmetric if M = M^t. Matrix M is anti-symmetric if M^t = -M....

n×n-matrix M is symmetric if M = M^t. Matrix M is anti-symmetric if M^t = -M.

1. Show that the diagonal of an anti-symmetric matrix are zero

2. suppose that A,B are symmetric n × n-matrices. Prove that AB is symmetric if AB = BA.

3. Let A be any n×n-matrix. Prove that A+A^t is symmetric and A - A^t antisymmetric.

4. Prove that every n × n-matrix can be written as the sum of a symmetric and anti-symmetric matrix.

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