## If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?

## Problem 647

Recall that a matrix $A$ is **symmetric** if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.

Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.

Otherwise, provide a counterexample.

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