---
tags: mth350, homework
---

# MTH 350 Homework 7

Instructions for this Homework set are on Blackboard where this is posted. The due date is on the class calendar, which is linked on Blackboard.

## Problem 1

:::warning
This problem is an **independent** problem. As described in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (top of page 14) this means that **on this problem, the only help you can get is from the professor.**
:::
Prove or disprove: The following subset of $M_2(\mathbb{R})$, with standard matrix addition and multiplication, is a field:
$$A = \left\{
\begin{pmatrix}
a & 0 \\ 0 & a
\end{pmatrix} \ : \ a \in \mathbb{R} \right\}$$

**Notes and hints:**

- **Before attempting this problem, make sure you are clear on the elements of this set**. It consists of all [diagonal matrices](https://en.wikipedia.org/wiki/Diagonal_matrix) where the entries on the main diagonal are equal. So for example $\left(\begin{smallmatrix}
2 & 0\\
0 & 2
\end{smallmatrix}\right) \in A$, but $\big(\begin{smallmatrix}
2 & 0\\
0 & 5
\end{smallmatrix}\big) \not \in A$ (because the diagonal entries are different), and $\big(\begin{smallmatrix}
2 & 0\\
1 & 2
\end{smallmatrix}\big) \not \in A$ (because one of the off-diagonal entries is nonzero).
- **Also before you attempt this problem, play with the set and the operations.** Choose several matrices in the set and add them, multiply them, try to find their additive and multiplicative inverses, and so on.

- You may assume the following about matrix multiplication and addition, since these are often proven in entry-level Linear Algebra courses:
- Addition is commutative
- Both operations are associative
- Matrix multiplication distributes over addition in both directions
- Those assumptions make it unnecessary to look at Axioms 1, 2, 5, 9, and 10 in the definition of a field. Your job is to focus on the remaining axioms. Are they all satisfied or not?
- Most of the remaining axioms have to do with the existence of something, for example "There exists an additive identity". To prove this, you'll need to demonstrate the matrix itself and then prove two things: (1) that it is actually an element of $A$, and (2) that it has the required property. For example if you believe $A$ has an additive identity, you'll need to state what specific matrix it is; then show that it's an element of $A$ (What does it mean to belong to $A$?), then show that when you add that matrix to any other matrix $M$ in $A$, you get $M$.
- Finally, although it's not listed in the axioms, you will also need to prove (or disprove) that $A$ is closed under addition and closed under multiplication. That is, given $M_1, M_2 \in A$, both $M_1 + M_2$ and $M_1M_2$ are in $A$. (What does this mean?)
- Be careful not to use "Proof by Assertion" --- i.e. when your proof consists just of stating the property (for example, "Multiplication is commutative because $M_1M_2 = M_2M_1$" with no other argument given). If you believe an axiom is satisfied, prove it, using elements of $A$. (But also beware not to just give examples.)
- If you believe you have a counterexample to one of the axioms, this (and an explanation of why it's a counterexample) is the only thing that needs to be turned in --- there's no need to state which axioms *are* satisfied.

## Problem 2
There is no problem 2 this time. :tada:

## Submission instructions

Just turn in your Problem 1 solution this time.

Remember to type up your work using $\LaTeX$.

Then upload your PDF to the "Homework 7" assignment area, and remember to hit **Submit**.

**Handy $\LaTeX$ commands:**

Here is the code that generated the definition of $A$ above:

A = \left\{ 
\begin{pmatrix} 
a & 0 \\ 
0 & a
\end{pmatrix} \ : \ a \in \mathbb{R} \right\}

The `pmatrix` environment handles the matrix itself. (The `p` stands for parenthesis; there's also a `bmatrix` environment that puts square brackets around the matrix if you like those better.) Note the `\left` and `\right` on the set brackets will resize those brackets to fit whatever is inside.

By default, `pmatrix` makes a matrix big. If you want it small, like in the first note above, use the `smallmatrix` environment:

\left(\begin{smallmatrix}
  2 & 0 \\
  0 & 2
\end{smallmatrix}\right)

Note the `\left` and `\right` again, this time used on parentheses. Those can be used on any delimiter to scale them up.