To show that if a matrix is skew symmetric, then its diagonal entries must all be 0, we can use the definition of a skew symmetric matrix.
A matrix A is skew symmetric if its transpose AT is equal to the negative of itself, -A.
Let's consider the diagonal entries of A. The diagonal entries of A are the entries where the row index is equal to the column index.
Now, if we take the transpose of A, the row indices become column indices and vice versa. So, the diagonal entries of AT will be the entries where the column index is equal to the row index.
Since A is skew symmetric, we have AT = -A. Therefore, the diagonal entries of AT must be equal to the corresponding diagonal entries of -A.
But since the diagonal entries of -A are the negation of the diagonal entries of A, we can conclude that the diagonal entries of A must be 0 in order for AT = -A to hold.
So, if a matrix is skew symmetric, its diagonal entries must all be 0.
