Matrix Multiplication Flashcards: Dimensions, Entries & Order
Practise 40 original matrix multiplication prompts: check dimensions, calculate row-by-column entries, compare AB and BA, and repair common errors.
Um þennan stokk
Check the dimensions before doing the arithmetic. These 40 matrix multiplication flashcards give English-reading learners original short problems on compatible shapes, row-by-column entries, small complete products, and the order of the factors.
You’ll decide whether a stated product exists and find its shape; calculate individual entries and short products with signed integers and zeros; and compare AB with BA. Selected reverse problems recover one missing dimension or entry from enough information to determine it uniquely. Local repairs target elementwise multiplication, a wrong row, a missed term, output dimensions, reversed factors, and a sign error. Every front supplies its own matrices or dimensions for mixed review.
The sequence begins with shape notation, compatibility, and the entry rule. It then introduces short calculations before order comparisons, constrained reverse problems, and error repair. Later cards revisit these decisions with rectangular matrices, cancellation, a zero column, a zero product from nonzero factors, and particular pairs that commute, including an identity matrix. Related calculation variants are separated by at least three other cards. All cards use the 'matrix-multiplication' tag; the review scheduler handles long-term spacing.
Know rows, columns, signed arithmetic, and simple equations before starting. Matrix indices start at 1, dimensions are positive integers, and a 1 × 1 product stays in brackets. This evergreen Mathematics deck has no exam or curriculum alignment. It excludes determinants, matrix inverses, elimination, eigenvalues, proofs, transformation composition, programming, and long 3 × 3 product drills. It does not mechanically reverse every calculation: arbitrary output matrices do not uniquely determine their inputs.
The compatibility and row-by-column rules were checked against Georgia Tech’s Interactive Linear Algebra. Only common mathematical facts are used. Every numerical example, prompt, answer, and arrangement is independently authored; no source exercises, textbook wording, competitor cards, or diagrams were copied or adapted.
AI tools assisted with drafting and the original decorative cover. Every final front and back was read in author review, and numerical answers were independently checked with exact integer arithmetic or substitution. This does not claim learner testing or guaranteed results. The CC0 dedication covers original wording, invented examples, organization, metadata, and generated cover to the extent applicable rights exist. Mathematical facts and cited materials retain their own status. This is independent study practice, with no affiliation to or endorsement by the cited organizations.
Spjöld í þessum stokki
Spjald 1
Spurning
What are the dimensions (rows × columns) of this matrix?
[ 6 -1 0 ] [ 2 3 5 ]Svar
2 × 3. There are two rows and three columns.
Spjald 2
Spurning
A is m × n and B is n × p. What are the dimensions of AB?
Svar
m × p. The shared inner dimension n makes the product defined; the outer dimensions give its shape. All dimensions here are positive integers.
Spjald 3
Spurning
For a defined matrix product C = AB, which parts of A and B determine entry C(i, j)? Indices start at 1.
Svar
Row i of A and column j of B. Multiply corresponding entries, then add every pairwise product.
Spjald 4
Spurning
A is 4 × 2 and B is 2 × 3. Is AB defined, and what is its shape?
Svar
Yes: 4 × 3. The inner dimensions are both 2.
Spjald 5
Spurning
Find entry (2, 1) of AB. Indices start at 1.
A = [ -1 4 ] [ 3 2 ] B = [ 5 -2 ] [ 1 6 ]Svar
17. Use row 2 of A and column 1 of B: 3 × 5 + 2 × 1 = 17.
Spjald 6
Spurning
A is 2 × 3 and B is 2 × 4. Is AB defined?
Svar
No. A has 3 columns, but B has 2 rows. The inner dimensions must match.
Spjald 7
Spurning
Calculate AB, keeping the result in matrix brackets.
A = [ 2 -1 3 ] B = [ 4 ] [ 5 ] [ -2 ]Svar
[ -3 ]The 1 × 1 entry is 2 × 4 + (−1) × 5 + 3 × (−2) = −3.
Spjald 8
Spurning
Calculate AB. Is the product undefined or a valid matrix?
A = [ 0 2 ] B = [ 3 ] [ 0 ]Svar
A valid 1 × 1 matrix:
[ 0 ]The dimensions match, and 0 × 3 + 2 × 0 = 0.
Spjald 9
Spurning
Find entry (1, 2) of AB. Indices start at 1.
A = [ 1 -2 4 ] [ 3 0 -1 ] B = [ 2 5 ] [ -1 3 ] [ 4 -2 ]Svar
−9. Row 1 meets column 2: 1 × 5 + (−2) × 3 + 4 × (−2) = 5 − 6 − 8 = −9.
Spjald 10
Spurning
A is 3 × k and B is 5 × 2. AB is defined. What is k?
Svar
k = 5. A must have as many columns as B has rows. AB then has shape 3 × 2.
Spjald 11
Spurning
Calculate AB.
A = [ 3 ] [ -2 ] B = [ 1 4 ]Svar
[ 3 12 ] [ -2 -8 ]A 2 × 1 matrix times a 1 × 2 matrix gives a 2 × 2 matrix. Each entry has one pairwise product.
Spjald 12
Spurning
A learner multiplies matching positions and reports entry (1, 1) of AB as 8. What should that entry be?
A = [ 2 1 ] [ 0 3 ] B = [ 4 -1 ] [ 5 2 ]Svar
13. Ordinary matrix multiplication uses row 1 against column 1: 2 × 4 + 1 × 5 = 13. Multiplying matching positions omits the second term.
Spjald 13
Spurning
A is 2 × 3 and B is 3 × 2. Give the shape of AB and the shape of BA, or say undefined.
Svar
AB: 2 × 2. BA: 3 × 3. Both inner-dimension checks pass, but reversing the order changes the output shape.
Spjald 14
Spurning
For AB below, a learner stops at 8 − 6 = 2. What term is missing, and what is the correct 1 × 1 result?
A = [ 2 -3 1 ] B = [ 4 ] [ 2 ] [ -5 ]Svar
The missing term is 1 × (−5) = −5; the result is [−3]. All three pairs contribute: 8 − 6 − 5 = −3.
Spjald 15
Spurning
Calculate AB.
A = [ 1 0 2 ] [ -1 3 1 ] B = [ 2 -1 ] [ 0 4 ] [ 3 2 ]Svar
[ 8 3 ] [ 1 15 ]The entries are 2 + 0 + 6, −1 + 0 + 4, −2 + 0 + 3, and 1 + 12 + 2.
Spjald 16
Spurning
For these matrices, compare entry (1, 2) of AB with entry (1, 2) of BA. Are the products equal?
A = [ 1 1 ] [ 0 1 ] B = [ 2 0 ] [ 0 3 ]Svar
AB has entry 3; BA has entry 2. The products are unequal. AB uses 1 × 0 + 1 × 3; BA uses 2 × 1 + 0 × 1. Both products are 2 × 2, so equal shapes do not imply equal matrices.
Spjald 17
Spurning
AB = [13]. Find x.
A = [ x 2 ] B = [ 3 ] [ -1 ]Svar
x = 5. The entry equation is 3x − 2 = 13. Substitution gives 3 × 5 − 2 = 13.
Spjald 18
Spurning
A learner uses row 1 of A and gets 4 for entry (2, 1) of AB. Correct the entry.
A = [ 2 -1 ] [ 4 3 ] B = [ 1 5 ] [ -2 0 ]Svar
−2. Entry (2, 1) needs row 2 of A: 4 × 1 + 3 × (−2) = −2.
Spjald 19
Spurning
A is 2 × 3 and B is 4 × 2. Give the shape of AB and BA, or say undefined.
Svar
AB: undefined. BA: 4 × 3. For AB, 3 ≠ 4. For BA, the inner dimensions are both 2.
Spjald 20
Spurning
Calculate AB.
A = [ -2 1 ] [ 3 0 ] B = [ 4 -1 ] [ 2 5 ]Svar
[ -6 7 ] [ 12 -3 ]Row 1 gives −8 + 2 and 2 + 5; row 2 gives 12 + 0 and −3 + 0.
Spjald 21
Spurning
A is m × 3, B is 3 × 4, and AB is 2 × 4. What is m?
Svar
m = 2. AB keeps the number of rows of the left matrix A.
Spjald 22
Spurning
Do AB and BA agree for this particular pair?
A = [ 2 0 ] [ 0 -1 ] B = [ 3 0 ] [ 0 4 ]Svar
Yes. Both products are:
[ 6 0 ] [ 0 -4 ]Some matrix pairs commute. Matrix multiplication is not commutative in general.
Spjald 23
Spurning
Find entry (3, 2) of AB. Indices start at 1.
A = [ 1 2 ] [ 0 -3 ] [ -2 4 ] B = [ 3 1 -1 ] [ 2 -2 5 ]Svar
−10. Use row 3 and column 2: (−2) × 1 + 4 × (−2) = −10.
Spjald 24
Spurning
A is 2 × 3 and B is 4 × 5. Is either AB or BA defined?
Svar
Neither is defined. AB fails because 3 ≠ 4; BA fails because 5 ≠ 2.
Spjald 25
Spurning
Entry (2, 1) of AB is 22. Find x.
A = [ 2 -1 ] [ 1 3 ] B = [ x 4 ] [ 5 -2 ]Svar
x = 7. Entry (2, 1) gives x + 3 × 5 = 22. Substitution gives 7 + 15 = 22.
Spjald 26
Spurning
Calculate both AB and BA.
A = [ 1 -2 ] B = [ 3 ] [ 4 ]Svar
AB:
[ -5 ]BA:
[ 3 -6 ] [ 4 -8 ]AB sums 3 − 8. BA is a 2 × 2 outer product.
Spjald 27
Spurning
Does AB equal the zero matrix for this pair, even though neither factor is the zero matrix?
A = [ 1 -1 ] [ 2 -2 ] B = [ 3 0 ] [ 3 0 ]Svar
Yes.
AB = [ 0 0 ] [ 0 0 ]The first column gives 3 − 3 and 6 − 6; the second gives zeros. A zero product does not force either factor to be the zero matrix.
Spjald 28
Spurning
A is 3 × 2 and B is 2 × 5. A learner calls AB a 2 × 2 matrix because the inner dimensions are 2. Correct the shape.
Svar
3 × 5. The matching inner dimensions allow multiplication; the outer dimensions determine the result.
Spjald 29
Spurning
Calculate AB.
A = [ 2 0 -1 ] B = [ 3 1 ] [ 4 -2 ] [ -2 5 ]Svar
[ 8 -3 ]The entries are 6 + 0 + 2 = 8 and 2 + 0 − 5 = −3. The result has shape 1 × 2.
Spjald 30
Spurning
A is 2 × 4 and B is 4 × 3. How many pairwise products must be added to calculate entry (2, 3) of AB?
Svar
Four. Row 2 of A and column 3 of B each contain four entries, so every output entry sums four pairwise products.
Spjald 31
Spurning
AB = [10]. Find x.
A = [ 2 x -1 ] B = [ 3 ] [ -2 ] [ 4 ]Svar
x = −4. The equation is 6 − 2x − 4 = 10, so −2x = 8. Substitution gives 6 + 8 − 4 = 10.
Spjald 32
Spurning
What is column 2 of AB?
A = [ 1 -2 4 ] [ 0 3 5 ] B = [ 2 0 ] [ -1 0 ] [ 6 0 ]Svar
[ 0 ] [ 0 ]Each entry in column 2 uses the all-zero column 2 of B, so every term is zero.
Spjald 33
Spurning
A learner computes BA while asked for AB and reports entry (1, 1) as 3. What is entry (1, 1) of AB?
A = [ 1 2 ] [ 0 -1 ] B = [ 3 0 ] [ 4 2 ]Svar
11. AB uses row 1 of A and column 1 of B: 1 × 3 + 2 × 4 = 11. The value 3 comes from reversing the factors.
Spjald 34
Spurning
A and B are both 2 × 3. Does having the same shape make AB defined?
Svar
No. The inner dimensions are 3 and 2, so AB is undefined. Equal shapes alone do not ensure compatibility.
Spjald 35
Spurning
Find entry (1, 2) of AB. Indices start at 1.
A = [ 2 1 ] [ -1 3 ] B = [ 4 -2 ] [ 2 4 ]Svar
0. The entry is 2 × (−2) + 1 × 4 = −4 + 4 = 0. Cancellation is a valid result.
Spjald 36
Spurning
A is 4 × 2 and B is k × 3. AB is defined and has shape 4 × 3. What is k?
Svar
k = 2. B must have two rows to match A’s two columns.
Spjald 37
Spurning
For AB below, a learner writes −2 + 12 = 10. Correct the sign error and give the 1 × 1 result.
A = [ 1 -3 ] B = [ -2 ] [ 4 ]Svar
[−14]. The second term is (−3) × 4 = −12, so the entry is −2 − 12 = −14.
Spjald 38
Spurning
Calculate AI and IA for these matrices.
A = [ 2 -1 ] [ 4 3 ] I = [ 1 0 ] [ 0 1 ]Svar
Both equal A:
[ 2 -1 ] [ 4 3 ]Multiplication by this 2 × 2 identity matrix preserves A in either order.
Spjald 39
Spurning
Find entry (3, 1) of AB. Indices start at 1.
A = [ 2 0 ] [ -1 4 ] [ 5 -2 ] B = [ 3 ] [ 7 ]Svar
1. Row 3 of A meets column 1 of B: 5 × 3 + (−2) × 7 = 15 − 14 = 1.
Spjald 40
Spurning
Calculate AB.
A = [ 3 -1 ] [ 2 4 ] B = [ 2 ] [ -3 ]Svar
[ 9 ] [ -8 ]The rows give 6 + 3 = 9 and 4 − 12 = −8. The result has shape 2 × 1.
40 spjöld
Matrix Multiplication Flashcards: Dimensions, Entries & Order
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