Teacher page: Matrices

The one thing to do with the animation

Ask for the top left entry of AB before stepping through.

Many will say 0, multiplying the two top-left entries. It is 2, because it uses the whole first row and the whole first column. Stepping through shows exactly which four numbers are involved.

Then do BA. Every product you have met commutes, so this one has to be shown rather than asserted, and it is the reason you cannot simply divide both sides of a matrix equation.

Check the orders first

Inside numbers match, outside numbers give the answer's size. A 2×3 times a 3×2 is a 2×2; a 2×3 times a 2×3 does not exist.

Making this the first line of every matrix question prevents the most common wasted minutes in the sub-topic, which are spent multiplying things that cannot be multiplied.

The answers

1. Top left of AB	1(0) + 2(1) = 2.
2. Top left of BA	0(1) + 1(3) = 3.
3. 2×3 times 3×2	A, a 2×2 matrix.

Where the marks go

1 markCompatible orders checked before multiplying.

1 markThe product taken in the order the question states.

1 markThe result interpreted in context when the matrices represent something.

What each wrong answer tells you

They give	What it means
0 (Q1)	Multiplied elementwise. The commonest first attempt, and worth naming rather than just correcting.
3 (Q1)	Computed BA. Order matters.
2 (Q2)	Computed AB.
B, 3×3 (Q3)	Reversed the product. Both exist here and they are different sizes, which is a useful thing to notice.
D, not defined (Q3)	Did not check the inner dimensions, which do match.

Other things they will say

"Why this strange rule?" Because a matrix represents a transformation, and the product represents doing one then the other. Doing them in the other order is a different thing, which is why AB and BA differ.

"Can I divide by a matrix?" No. There is an inverse, and you must choose which side to multiply by, which is precisely because order matters.

"Does the calculator do this?" Yes, and it should for anything above 2×2. Do one by hand so the calculator's answer means something.

A possible order

	What is happening
1	Predict the top-left entry. Step through.
2	Orders and when a product exists.
3	Addition, scalars, and 2×2 products by hand.
4	AB against BA, deliberately.
5	A context: costs times quantities, and what the answer means.

Two things not to say

Do not say "multiply the matrices" without specifying the order. They will pick one at random.

Do not move to the calculator before one product has been done by hand.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

MultiplyA = [[2,0],[1,3]], B = [[1,4],[2,1]]. Find AB.
[[2,8],[7,7]]
DeterminantFind det[[3,5],[2,4]].
2

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Use in contextCosts per item are [40, 65] baht. Orders from two shops are [[12, 5],[8, 9]]. Compute the matrix product that gives each shop's total and interpret it.
[[805],[905]] baht. Row by column: quantity times price, summed.
Order checkA is 3×2 and B is 2×4. Which of AB and BA exists, and what size is it?
AB exists and is 3×4. BA does not: the inner dimensions 4 and 3 do not match.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Find an unknown entryA = [[1,2],[3,k]]. Find k so that det A = 0, and say what that means.
k = 6. The rows become multiples of each other, so the matrix squashes the plane onto a line and has no inverse.
Reason about orderGive two 2×2 matrices with AB = BA, and explain why your example works when most do not.
Any matrix with the identity, or two matrices that are both multiples of the same one. Commuting needs the two transformations to leave each other's directions alone.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.