Matrix Multiplication

Doing one transformation, then another.

12 problems · about 24 minutes · N-VM.C.8, N-VM.C.9

What this lesson teaches

The student multiplies matrices and explains why the operation is not commutative.

AB means apply B first, then A, so matrix multiplication is composition.
The entry in row i, column j is row i of A dotted with column j of B.
AB and BA are usually different, because doing two things in the other order gives another result.

Straightforward practice. Get the method working first.

5 problems

A 5 by 2 times a 2 by 3. How many columns does the product have?
Answer 3
Why 3.
What does the product AB mean?
Answer Apply B first, then apply A to the result.
Why Composition, right to left.
Row 1 of A is [2, 0] and column 1 of B is ⟨3, 7⟩. What is the entry in row 1, column 1 of AB?
Answer 6
Why 6 + 0.
A 3 by 2 times a 2 by 4. How many rows does the product have?
Answer 3
Why The first outer number.
A 3 by 2 times a 4 by 2. Is the product defined? 1 yes, 0 no.
Answer 0
Why The inner numbers 2 and 4 disagree.

The same ideas with more to keep track of.

3 problems

Is AB always equal to BA? 1 yes, 0 no.
Answer 0
Why Order changes the result.
In AB, which matrix acts on a vector first? Enter 1 for A, 2 for B.
Answer 2
Why Read right to left.
The 3 by 3 identity matrix. How many ones does it contain?
Answer 3
Why One per diagonal position.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps for computing one entry of a matrix product.
Answer 1. Check the inner dimensions agree 2. Take the row of A matching the entry position 3. Take the column of B matching the entry position 4. Dot them together
Why Check the sizes before doing any arithmetic.
A 4 by 4 matrix times a 4 by 4 matrix. How many entries does the product have?
Answer 16
Why 4 by 4.
The Shape: A 2 by 3 matrix times a 3 by 5 matrix. How many columns does the product have?
Answer 5
Why 5.
The Identity: A times the identity matrix. Is the result A? 1 yes, 0 no.
Answer 1
Why Yes.