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Cogito

Linear Algebra · Chapter 3 · Lesson 2

Practice Worksheet

Matrix Multiplication

Doing one transformation, then another.

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

NameDate

Figure — use these to answer the problems

Set a shear, then a rotation, and note where the arrows land. Swap the order and compare: the results differ.
ijv
  • i-hat lands on(1, 0)
  • j-hat lands on(1, 1)
  • Determinant1
  • v lands on(3, 1)
a1
b1
c0
d1
v, first entry2
v, second entry1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A 5 by 2 times a 2 by 3. How many columns does the product have?

    Answer

  2. What does the product AB mean?

    • Apply B first, then apply A to the result.
    • Multiply matching entries together.

  3. 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

  4. A 3 by 2 times a 2 by 4. How many rows does the product have?

    Answer

  5. A 3 by 2 times a 4 by 2. Is the product defined? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Is AB always equal to BA? 1 yes, 0 no.

    Answer

  2. In AB, which matrix acts on a vector first? Enter 1 for A, 2 for B.

    Answer

  3. The 3 by 3 identity matrix. How many ones does it contain?

    Answer

Stretch Yourself

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

4 problems

  1. Order the steps for computing one entry of a matrix product.

    Write 1 to 4 in the boxes to put these in order.

    • Check the inner dimensions agree
    • Take the row of A matching the entry position
    • Take the column of B matching the entry position
    • Dot them together

  2. A 4 by 4 matrix times a 4 by 4 matrix. How many entries does the product have?

    Answer

  3. The Shape

    A 2 by 3 matrix times a 3 by 5 matrix. How many columns does the product have?

    Answer

  4. The Identity

    A times the identity matrix. Is the result A? 1 yes, 0 no.

    Answer