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

symmetric matrices problems

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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Proof.

This is true.

If $A$ is in reduced row echelon form, then every term below the diagonal must be 0.
That is, the entry $a_{i j} = 0$ for all $i > j$.

If $A$ is, additionally, symmetric, then for $i < j$ we also have $a_{j i} = a_{i j} = 0$.

These two facts together means that $a_{i j} = 0$ whenever $i \neq j$.

In this case, the only non-zero terms in the matrix $A$ lie on the diagonal, and so $A$ is a diagonal matrix.


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