Matrix Multiplication Practice: Dimensions, Entries, and Order
A product of two matrices can have the right dimensions and still contain the wrong numbers. Checking the shape catches one kind of mistake. Checking which row and column produced an entry catches another.
This matrix multiplication practice worksheet separates those checks. Try seven original exercises covering rectangular products, an order that doesn't exist, and square products whose shapes match but whose entries disagree. The worked answers show every requested calculation. Use paper so you can keep the row and column selections beside your arithmetic.
Check dimensions before calculating
A matrix's dimensions are written rows × columns. For AB, the number of columns in A must equal the number of rows in B. The answer keeps the number of rows from A and the number of columns from B:
A B AB
m x n n x p m x p
^ ^
These must match.
Each entry of AB comes from one row of A and one column of B: multiply corresponding numbers, then add. Entry (i, j) uses row i and column j, counting from 1. Georgia Tech's Interactive Linear Algebra chapter on matrix multiplication explains this row-column rule and how matrix products represent combinations of linear transformations.
Before doing arithmetic, draw an empty answer with its predicted shape. For a 2 × 3 answer, that means two rows with three spaces each. Fill one row at a time: keep the same row of the left matrix while moving across the columns of the right matrix.
One worked example
Calculate WX:
W = [ 2 -1 3 ] X = [ 1 2 ]
[ 0 4 1 ] [ 3 0 ]
[ -2 5 ]
The dimensions are (2 × 3)(3 × 2), so WX is 2 × 2. Its top-right entry uses row 1 of W and column 2 of X:
(2)(2) + (-1)(0) + (3)(5) = 4 + 0 + 15 = 19
Calculate the other three positions in the same way:
(1,1): (2)(1) + (-1)(3) + (3)(-2) = -7
(2,1): (0)(1) + (4)(3) + (1)(-2) = 10
(2,2): (0)(2) + (4)(0) + (1)(5) = 5
WX = [ -7 19 ]
[ 10 5 ]
Each entry has three multiplication terms because the selected row and column each contain three numbers. If you've written only two terms, check for a missing pair.
Seven matrix multiplication exercises
Try these before reading the answers. Write the dimension check for every requested product. If a product doesn't exist, identify the dimensions that fail to match. Each exercise defines its own matrices; A and B can change between questions.
1. Predict the shape
For each pair, decide whether AB and BA exist. Give the dimensions of every product that exists. No entries are needed.
| Pair | Dimensions of A | Dimensions of B |
|---|---|---|
| a | 2 × 3 |
3 × 4 |
| b | 3 × 2 |
2 × 3 |
| c | 2 × 2 |
2 × 2 |
2. Find two entries without doing everything
Let C = AB. Find entries (2, 3) and (3, 2) of C. Write the chosen row and column for each.
A = [ 1 2 ] B = [ 2 0 5 ]
[ -3 4 ] [ 1 -2 3 ]
[ 0 -1 ]
3. Compute a rectangular product
Find AB. Then give the dimensions of BA without calculating its entries.
A = [ 1 2 -1 ] B = [ 2 1 ]
[ 3 0 2 ] [ -1 4 ]
[ 3 -2 ]
4. Only one order works
Find Cd. Does dC exist?
C = [ 2 0 -1 ] d = [ 4 ]
[ 1 3 2 ] [ -2 ]
[ 5 ]
5. Both orders work, with different shapes
Find rs and sr. Keep brackets around the 1 × 1 matrix.
r = [ 1 -2 3 ] s = [ 2 ]
[ 0 ]
[ -1 ]
6. Same shape, different answer
Calculate AB and BA. Do these particular matrices commute, meaning AB = BA?
A = [ 1 2 ] B = [ 2 0 ]
[ 0 1 ] [ 3 1 ]
7. Diagnose a shape-correct answer
Someone multiplies numbers in matching positions and reports R as PQ. Is the answer correct? Repair all four entries.
P = [ 2 -1 ] Q = [ 4 2 ]
[ 3 0 ] [ -2 5 ]
R = [ 8 -2 ]
[ -6 0 ]
Answers with the calculations visible
1. Reverse the dimension check too
For pair a, AB exists: (2 × 3)(3 × 4) gives 2 × 4. But BA would be (3 × 4)(2 × 3). Its inner dimensions, 4 and 2, don't match, so BA is undefined.
For pair b, both products exist:
AB: (3 x 2)(2 x 3) gives 3 x 3
BA: (2 x 3)(3 x 2) gives 2 x 2
For pair c, both products are 2 × 2 because both checks are (2 × 2)(2 × 2). This establishes that the multiplications are allowed. Without entries, it doesn't establish that their results are equal.
2. The position chooses the inputs
AB is 3 × 3. Entry (2, 3) uses row 2 of A, [-3, 4], and column 3 of B, [5, 3]. Entry (3, 2) uses row 3, [0, -1], and column 2, [0, -2]:
(2,3): (-3)(5) + (4)(3) = -15 + 12 = -3
(3,2): (0)(0) + (-1)(-2) = 0 + 2 = 2
The row number comes first. Reversing the position numbers selects different inputs and can change the answer.
3. Four entries, three pairs in each
The dimension check is (2 × 3)(3 × 2), so AB is 2 × 2:
(1,1): (1)(2) + (2)(-1) + (-1)(3) = -3
(1,2): (1)(1) + (2)(4) + (-1)(-2) = 11
(2,1): (3)(2) + (0)(-1) + (2)(3) = 12
(2,2): (3)(1) + (0)(4) + (2)(-2) = -1
AB = [ -3 11 ]
[ 12 -1 ]
BA also exists: (3 × 2)(2 × 3) gives 3 × 3. Its different dimensions already rule out equality with AB.
4. A failed product has no entries to calculate
Cd has dimensions (2 × 3)(3 × 1), giving 2 × 1:
(1,1): (2)(4) + (0)(-2) + (-1)(5) = 3
(2,1): (1)(4) + (3)(-2) + (2)(5) = 8
Cd = [ 3 ]
[ 8 ]
dC would have dimensions (3 × 1)(2 × 3). The inner dimensions are 1 and 2, so it is undefined. Writing a zero matrix would be a different answer: zero entries can occur in a valid product, but this multiplication isn't defined.
5. A single entry versus nine entries
For rs, the dimensions are (1 × 3)(3 × 1), giving 1 × 1:
rs = [ (1)(2) + (-2)(0) + (3)(-1) ] = [ -1 ]
For sr, the dimensions are (3 × 1)(1 × 3), giving 3 × 3. Each entry has just one multiplication term:
sr = [ (2)(1) (2)(-2) (2)(3) ]
[ (0)(1) (0)(-2) (0)(3) ]
[ (-1)(1) (-1)(-2) (-1)(3) ]
= [ 2 -4 6 ]
[ 0 0 0 ]
[ -1 2 -3 ]
The same six input numbers produce two differently shaped matrices. Order controls which rows meet which columns.
6. Equal dimensions don't guarantee equal products
Both dimension checks are (2 × 2)(2 × 2). For AB:
(1,1): (1)(2) + (2)(3) = 8
(1,2): (1)(0) + (2)(1) = 2
(2,1): (0)(2) + (1)(3) = 3
(2,2): (0)(0) + (1)(1) = 1
AB = [ 8 2 ]
[ 3 1 ]
Reverse the inputs for BA:
(1,1): (2)(1) + (0)(0) = 2
(1,2): (2)(2) + (0)(1) = 4
(2,1): (3)(1) + (1)(0) = 3
(2,2): (3)(2) + (1)(1) = 7
BA = [ 2 4 ]
[ 3 7 ]
The top-left entries, 8 and 2, already disprove equality. These matrices don't commute. Matrix multiplication is not commutative in general; some particular pairs do commute. “The products are always different” would be too strong.
7. The shape passes; the method fails
R is the elementwise product: each entry multiplies only the numbers in the same position. Ordinary matrix multiplication combines a complete row and column:
(1,1): (2)(4) + (-1)(-2) = 10
(1,2): (2)(2) + (-1)(5) = -1
(2,1): (3)(4) + (0)(-2) = 12
(2,2): (3)(2) + (0)(5) = 6
PQ = [ 10 -1 ]
[ 12 6 ]
Both R and PQ are 2 × 2, so a shape check can't distinguish them. Recomputing the top-left entry reveals the wrong operation.
Review the decision you missed
Label each miss: dimensions, row/column selection, arithmetic, reversed order, or elementwise multiplication. Retry that exercise on blank paper.
For a short recall prompt, ask: “A is 2 × 3 and B is 3 × 4. Does BA exist?” Answer: “No. Its inner dimensions are 4 and 2.” Another useful prompt is: “What produces entry (2, 3) of AB?” Answer: “Row 2 of A multiplied by column 3 of B, pairing corresponding entries and adding their products.” Apply that answer to exercise 2 with the numbers visible.
The math flashcards guide explains how to keep these reusable decisions separate from full solutions. Keep full matrix products in your practice too: knowing the rule and carrying out every entry are different tasks.
This worksheet was drafted and edited with AI assistance; the illustration was AI-generated. Its numerical products were checked with Python calculations.