Parker Glynn-Adey

Question 4 and Question 6

Posted in Uncategorized by pgadey on 2016/02/12

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

Question 4 How many distinct entries can {A} have if {A} is a {n \times n} 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,

\displaystyle A = \left( \begin{matrix} 1 & 2 \\ 3 & 4\\ \end{matrix} \right) \quad B = \left( \begin{matrix} 1 & 1 \\ 3 & 4\\ \end{matrix} \right) \quad

{A} has distinct entries and {B} does not have distinct entries.

Now, you need to know the definition of being symmetric: {M} symmetric if all {M = M^{T}}.

For example,

\displaystyle D = \left( \begin{matrix} 1 & 2 \\ 2 & 3\\ \end{matrix} \right) \quad E = \left( \begin{matrix} 4 & 5 \\ 6 & 7\\ \end{matrix} \right) \quad

{D} is symmetric and {E} 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 {3 \times 3} symmetric matrix with entries from {\{1,2,3,4,5,6\}}.

Or — Think of it this way.

Suppose that I give you:

\displaystyle M = \left( \begin{matrix} 1 & 2 & 3\\ x & 4 & 5\\ y & z & 6 \end{matrix} \right)

and tell you, “{M} is symmetric”. What are {x}, {y}, and {z}?

Question 6 Suppose that {A} is {n \times n} and that {AX = B} has a solution for every {B}. Show that {A} is invertible.

Consider what happens with {2 \times 2} matrices.

\displaystyle A = \left(\begin{matrix} a & b \\ c & d \end{matrix}\right)

Now suppose we can solve:

\displaystyle A \left(\begin{matrix} x_1\\ y_1\\ \end{matrix}\right) = \left(\begin{matrix} 1\\ 0 \end{matrix}\right)

and

\displaystyle A \left(\begin{matrix} x_2\\ y_2\\ \end{matrix}\right) = \left(\begin{matrix} 0\\ 1\\ \end{matrix}\right)

How can we find {p,q,r,s} so that?

\displaystyle A \left(\begin{matrix} p & q\\ r & s \end{matrix}\right) = \left(\begin{matrix} 1 & 0\\ 0 & 1\\ \end{matrix}\right)

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