## Question 4 and Question 6

These questions kept coming up during my MAT A33 office hours. The notes below are an extended set of hints for these questions.

Question 4How many distinct entries can have if is a symmetric matrix,

First you need to know the definition of *having distinct entries*: A matrix has distinct entries if all of its entries are different.

For example,

has distinct entries and does not have distinct entries.

Now, you need to know the definition of *being symmetric*: symmetric if all .

For example,

is symmetric and is not symmetric.

So — Essentially, the question asks “how many distinct entries can a symmetric matrix have?”

To get started working on the problem make a symmetric matrix with entries from .

Or — Think of it this way.

Suppose that I give you:

and tell you, “ is symmetric”. What are , , and ?

Question 6Suppose that is and that has a solution for every . Show that is invertible.

Consider what happens with matrices.

Now suppose we can solve:

and

How can we find so that?

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