Home » Uncategorized » The matrix is symmetric

# The matrix is symmetric

Let be a square matrix and for that it holds that

Prove that is symmetric.

Solution

Let be an square matrix over a field such that

(1)

Taking transposed matrices back at we get that

and thus

(2)

On the other hand it holds that

since

Of course it holds that if a matrix is symmetric or antisymmetric and then . The proof is left as an exercise to the reader.

We can safely conclude using the above observations that for our matrix it holds that

(3)

But then for the matrix it holds that

and since the matrix is antisymmetric we conclude that and thus . Hence the result.

### Who is Tolaso?

Find out more at his Encyclopedia Page.