Matrix Multiplication Properties

Lesson 9 of 12: VM Cluster

In this lesson:

• Demonstrate that matrix multiplication is not commutative:
• Verify associativity and distributive properties still hold
• Recognize zero divisors and order-dependent expansion failures

Recall: Arithmetic Properties for Real Numbers

• Commutative: — order never matters for numbers
• Associative: — grouping never matters
• Distributive:

Which of these do you expect matrices to inherit — and which might break?

Does Order Matter in Multiplication?

For numbers: — always.

For matrices — does swapping order give the same result?

Compute and , then compare.

Compute the Products and Compare

— one counterexample disproves commutativity entirely.

Order Failure: When Only One Product Exists

Sometimes only one order is defined:

• is , is
• : inner → defined ( result)
• : inner → undefined

Never assume . Always check order.

Why Non-Commutativity Is Geometrically Real

Transformations don't commute in the physical world.

Rotate a shape 90°, then reflect it across the x-axis.

Reflect the same shape across the x-axis, then rotate 90°.

The results are different orientations.

Associativity: You Can Move Parentheses

Even though commutativity fails, associativity holds:

• Parentheses can change — grouping is flexible
• The order of letters cannot change

✗ Wrong: — that's rearranging, not regrouping

Verifying Associativity with Specific Numbers

Using , , as matrices:

• Path 1:
• Path 2:

Both paths match — associativity confirmed. ✓

Two Distributive Laws: Left and Right

Left:

Right:

Because , these are two separate laws. You cannot write — that flips to the wrong side.

Verifying Distributivity with Specific Numbers

For specific matrices , , :

• ✓

Quick Check: Which Must Be True?

For any matrices where products are defined:

Which are always true?

Quick Check: Which Statements Always Hold

1. — counterexample exists
2. — associativity
3. — flips to wrong side
4. — right distributivity

Binomial Expansion Breaks in Matrix Algebra

Scalars:

Matrices:

Since , the middle terms cannot combine.

Find the Flaw in This Expansion

A student wrote:

The step from line 1 to line 2 is wrong.

What assumption did the student make? Where exactly?

Zero Divisors: AB Can Equal Zero

Scalars: implies or .

Matrices: this fails.

and — compute .

Computing the Zero-Divisor Product Directly

, , but . Zero-product property fails.

What Cancellation Rules Break Down

Since does not imply or :

• does not imply (needs invertible )
• — order cannot be rearranged

Cancellation requires explicit justification in matrix algebra.

Expand and Simplify These Expressions

Simplify, keeping careful track of order. No scaffolding — use what you know.

Hint: which property lets you expand each one? Where must you be careful about order?

Answers: Simplifying Three Matrix Expressions

1. in general

3. (scalar multiples commute with matrix factors)

Predict: Does BA Also Equal Zero?

For the zero-divisor pair:

Predict: Does as well? Then compute and check.

What Scalar Algebra Keeps in Matrices

Matrix Algebra: A Different Algebraic World

Matrix algebra is not scalar algebra with bigger symbols:

• Order matters — non-commutativity is structural
• ; zero products need no zero factor
• Cancellation requires explicit justification

Next: Does a "do-nothing" matrix exist? Does every matrix have an inverse? VM.C.10 answers both.