Question 376126: Question:
Matrix A is said to be skew symmetric if A^T = -A. Show that if a matrix is skew symmetric, then its diagonal entries must be 0.
Thanks in advance.
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Consider the matrix A = [a(j,k)], where a(j,k) is the entry in the jth row and kth column. It must follow that A must be a square matrix. The diagonal entries of A are a(1,1), a(2,2), a(3,3), ..., a(n,n). If we take the transpose of A, the diagonal entries remain fixed. Now  , and the diagonal entries of -A are -a(1,1), -a(2,2), -a(3,3), ..., -a(n,n).
Thus a(1,1) = -a(1,1), a(2,2) = -a(2,2), a(3,3) = -a(3,3), ...,
a(n,n) = -a(n,n). These imply that a(1,1) = 0, a(2,2) = 0, a(3,3) = 0, ...,
a(n,n) = 0, and the proof is complete.
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