In this lesson:
Which of these do you expect matrices to inherit — and which might break?
For numbers: — always.
For matrices — does swapping order give the same result?
Compute and , then compare.
— one counterexample disproves commutativity entirely.
Sometimes only one order is defined:
Never assume . Always check order.
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.
Even though commutativity fails, associativity holds:
✗ Wrong: — that's rearranging, not regrouping
Using , , as matrices:
Both paths match — associativity confirmed. ✓
Left:
Right:
Because , these are two separate laws. You cannot write — that flips to the wrong side.
For specific matrices , , :
For any matrices where products are defined:
Which are always true?
Scalars:
Matrices:
Since , the middle terms cannot combine.
A student wrote:
The step from line 1 to line 2 is wrong.
What assumption did the student make? Where exactly?
Scalars: implies or .
Matrices: this fails.
and — compute .
, , but . Zero-product property fails.
Since does not imply or :
Cancellation requires explicit justification in matrix algebra.
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?
in general
(scalar multiples commute with matrix factors)
For the zero-divisor pair:
Predict: Does as well? Then compute and check.
Matrix algebra is not scalar algebra with bigger symbols:
Next: Does a "do-nothing" matrix exist? Does every matrix have an inverse? VM.C.10 answers both.
Click to begin the narrated lesson
Understand matrix multiplication properties