Looking Forwards:

**Why are the rules of arithmetic important **

beyond high school mathematics ?

**Content knowledge:** Matrix arithmetic

**Goal:** Exploring why the rules of arithmetic are

important in linear algebra will deepen one's

perspective on the importance of the rules of

arithmetic.

** Number Systems **

There are some standard number systems:

Q = {p=q where p, q are integers and q ≠
0}

R is the set of real numbers .

C is the set of complex numbers .

There's many more!
or and
so on.

These aren't just sets. I want to think of each as a set together

with some arithmetic operations . For simplicity , we'll just consider

+ and · (and ignore subtraction and division ).

**Matrices**

The system of 2 ·2 matrices is generalized number system. Here

are some examples of 2 ·2 matrices:

or
, or

A general 2 ·2 matrix can be written as follows:

where a, b, c and d are real numbers.

Two matrices are equal if and only if every entry is

equal.

For instance:

means that

a = 1, b = 3, c = -4 and d = 2

There are NO values of x such that the following equation
is

true:

** Adding matrices **

You add matrices entrywise. For example:

Matrix addition can be summarized by the following
formula:

Matrix multiplication

Matrix multiplication can be summarized by the following

formula :

**Motivating Question:** What are the rules of

matrix arithmetic?

More specifically, which of the following statements are
true for all

2 ·2 matrices A, B and C:

1. (Commutative property of addition )

2. (Associative property of addition)

3. (Commutative property of multiplication)

4. ( Distributive property )

5. and so on . . .

Additive identity for numbers

If a is any number then:

a + 0 = a:

**Additive Identity for Matrices?**

True or False: If A is any 2 ·2 matrix, then:

If true, explain why. If false, provide a counterexample.

**Answer:** True.

**Moral:** is the additive identity for
the system of 2· 2

matrices.

** Multiplicative Identity for Matrices?**

**True or False:** If A is any 2 ·2 matrix, then:

If true, explain why. If false, provide a counterexample.

**Answer:** False.

**Multiplicative Identity for Matrices?**

**True or False: **If A is any 2 ·2 matrix, then:

If true, explain why. If false, provide a counterexample.

**Answer:** True.

**Moral:**is the multiplicative identity
for the system of

2 ·2 matrices.

What are the other rules of matrix arithmetic?