## 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?

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

Tagged with: ,