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.
เกี่ยวกับชุดการ์ดนี้
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.
การ์ดในชุดการ์ดนี้
การ์ด 1
คำถาม
What are the dimensions (rows × columns) of this matrix?
[ 6 -1 0 ] [ 2 3 5 ]คำตอบ
2 × 3. There are two rows and three columns.
การ์ด 2
คำถาม
A is m × n and B is n × p. What are the dimensions of AB?
คำตอบ
m × p. The shared inner dimension n makes the product defined; the outer dimensions give its shape. All dimensions here are positive integers.
การ์ด 3
คำถาม
For a defined matrix product C = AB, which parts of A and B determine entry C(i, j)? Indices start at 1.
คำตอบ
Row i of A and column j of B. Multiply corresponding entries, then add every pairwise product.
การ์ด 4
คำถาม
A is 4 × 2 and B is 2 × 3. Is AB defined, and what is its shape?
คำตอบ
Yes: 4 × 3. The inner dimensions are both 2.
การ์ด 5
คำถาม
Find entry (2, 1) of AB. Indices start at 1.
A = [ -1 4 ] [ 3 2 ] B = [ 5 -2 ] [ 1 6 ]คำตอบ
17. Use row 2 of A and column 1 of B: 3 × 5 + 2 × 1 = 17.
การ์ด 6
คำถาม
A is 2 × 3 and B is 2 × 4. Is AB defined?
คำตอบ
No. A has 3 columns, but B has 2 rows. The inner dimensions must match.
การ์ด 7
คำถาม
Calculate AB, keeping the result in matrix brackets.
A = [ 2 -1 3 ] B = [ 4 ] [ 5 ] [ -2 ]คำตอบ
[ -3 ]The 1 × 1 entry is 2 × 4 + (−1) × 5 + 3 × (−2) = −3.
การ์ด 8
คำถาม
Calculate AB. Is the product undefined or a valid matrix?
A = [ 0 2 ] B = [ 3 ] [ 0 ]คำตอบ
A valid 1 × 1 matrix:
[ 0 ]The dimensions match, and 0 × 3 + 2 × 0 = 0.
การ์ด 9
คำถาม
Find entry (1, 2) of AB. Indices start at 1.
A = [ 1 -2 4 ] [ 3 0 -1 ] B = [ 2 5 ] [ -1 3 ] [ 4 -2 ]คำตอบ
−9. Row 1 meets column 2: 1 × 5 + (−2) × 3 + 4 × (−2) = 5 − 6 − 8 = −9.
การ์ด 10
คำถาม
A is 3 × k and B is 5 × 2. AB is defined. What is k?
คำตอบ
k = 5. A must have as many columns as B has rows. AB then has shape 3 × 2.
การ์ด 11
คำถาม
Calculate AB.
A = [ 3 ] [ -2 ] B = [ 1 4 ]คำตอบ
[ 3 12 ] [ -2 -8 ]A 2 × 1 matrix times a 1 × 2 matrix gives a 2 × 2 matrix. Each entry has one pairwise product.
การ์ด 12
คำถาม
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 ]คำตอบ
13. Ordinary matrix multiplication uses row 1 against column 1: 2 × 4 + 1 × 5 = 13. Multiplying matching positions omits the second term.
การ์ด 13
คำถาม
A is 2 × 3 and B is 3 × 2. Give the shape of AB and the shape of BA, or say undefined.
คำตอบ
AB: 2 × 2. BA: 3 × 3. Both inner-dimension checks pass, but reversing the order changes the output shape.
การ์ด 14
คำถาม
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 ]คำตอบ
The missing term is 1 × (−5) = −5; the result is [−3]. All three pairs contribute: 8 − 6 − 5 = −3.
การ์ด 15
คำถาม
Calculate AB.
A = [ 1 0 2 ] [ -1 3 1 ] B = [ 2 -1 ] [ 0 4 ] [ 3 2 ]คำตอบ
[ 8 3 ] [ 1 15 ]The entries are 2 + 0 + 6, −1 + 0 + 4, −2 + 0 + 3, and 1 + 12 + 2.
การ์ด 16
คำถาม
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 ]คำตอบ
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.
การ์ด 17
คำถาม
AB = [13]. Find x.
A = [ x 2 ] B = [ 3 ] [ -1 ]คำตอบ
x = 5. The entry equation is 3x − 2 = 13. Substitution gives 3 × 5 − 2 = 13.
การ์ด 18
คำถาม
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 ]คำตอบ
−2. Entry (2, 1) needs row 2 of A: 4 × 1 + 3 × (−2) = −2.
การ์ด 19
คำถาม
A is 2 × 3 and B is 4 × 2. Give the shape of AB and BA, or say undefined.
คำตอบ
AB: undefined. BA: 4 × 3. For AB, 3 ≠ 4. For BA, the inner dimensions are both 2.
การ์ด 20
คำถาม
Calculate AB.
A = [ -2 1 ] [ 3 0 ] B = [ 4 -1 ] [ 2 5 ]คำตอบ
[ -6 7 ] [ 12 -3 ]Row 1 gives −8 + 2 and 2 + 5; row 2 gives 12 + 0 and −3 + 0.
การ์ด 21
คำถาม
A is m × 3, B is 3 × 4, and AB is 2 × 4. What is m?
คำตอบ
m = 2. AB keeps the number of rows of the left matrix A.
การ์ด 22
คำถาม
Do AB and BA agree for this particular pair?
A = [ 2 0 ] [ 0 -1 ] B = [ 3 0 ] [ 0 4 ]คำตอบ
Yes. Both products are:
[ 6 0 ] [ 0 -4 ]Some matrix pairs commute. Matrix multiplication is not commutative in general.
การ์ด 23
คำถาม
Find entry (3, 2) of AB. Indices start at 1.
A = [ 1 2 ] [ 0 -3 ] [ -2 4 ] B = [ 3 1 -1 ] [ 2 -2 5 ]คำตอบ
−10. Use row 3 and column 2: (−2) × 1 + 4 × (−2) = −10.
การ์ด 24
คำถาม
A is 2 × 3 and B is 4 × 5. Is either AB or BA defined?
คำตอบ
Neither is defined. AB fails because 3 ≠ 4; BA fails because 5 ≠ 2.
การ์ด 25
คำถาม
Entry (2, 1) of AB is 22. Find x.
A = [ 2 -1 ] [ 1 3 ] B = [ x 4 ] [ 5 -2 ]คำตอบ
x = 7. Entry (2, 1) gives x + 3 × 5 = 22. Substitution gives 7 + 15 = 22.
การ์ด 26
คำถาม
Calculate both AB and BA.
A = [ 1 -2 ] B = [ 3 ] [ 4 ]คำตอบ
AB:
[ -5 ]BA:
[ 3 -6 ] [ 4 -8 ]AB sums 3 − 8. BA is a 2 × 2 outer product.
การ์ด 27
คำถาม
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 ]คำตอบ
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.
การ์ด 28
คำถาม
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.
คำตอบ
3 × 5. The matching inner dimensions allow multiplication; the outer dimensions determine the result.
การ์ด 29
คำถาม
Calculate AB.
A = [ 2 0 -1 ] B = [ 3 1 ] [ 4 -2 ] [ -2 5 ]คำตอบ
[ 8 -3 ]The entries are 6 + 0 + 2 = 8 and 2 + 0 − 5 = −3. The result has shape 1 × 2.
การ์ด 30
คำถาม
A is 2 × 4 and B is 4 × 3. How many pairwise products must be added to calculate entry (2, 3) of AB?
คำตอบ
Four. Row 2 of A and column 3 of B each contain four entries, so every output entry sums four pairwise products.
การ์ด 31
คำถาม
AB = [10]. Find x.
A = [ 2 x -1 ] B = [ 3 ] [ -2 ] [ 4 ]คำตอบ
x = −4. The equation is 6 − 2x − 4 = 10, so −2x = 8. Substitution gives 6 + 8 − 4 = 10.
การ์ด 32
คำถาม
What is column 2 of AB?
A = [ 1 -2 4 ] [ 0 3 5 ] B = [ 2 0 ] [ -1 0 ] [ 6 0 ]คำตอบ
[ 0 ] [ 0 ]Each entry in column 2 uses the all-zero column 2 of B, so every term is zero.
การ์ด 33
คำถาม
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 ]คำตอบ
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.
การ์ด 34
คำถาม
A and B are both 2 × 3. Does having the same shape make AB defined?
คำตอบ
No. The inner dimensions are 3 and 2, so AB is undefined. Equal shapes alone do not ensure compatibility.
การ์ด 35
คำถาม
Find entry (1, 2) of AB. Indices start at 1.
A = [ 2 1 ] [ -1 3 ] B = [ 4 -2 ] [ 2 4 ]คำตอบ
0. The entry is 2 × (−2) + 1 × 4 = −4 + 4 = 0. Cancellation is a valid result.
การ์ด 36
คำถาม
A is 4 × 2 and B is k × 3. AB is defined and has shape 4 × 3. What is k?
คำตอบ
k = 2. B must have two rows to match A’s two columns.
การ์ด 37
คำถาม
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 ]คำตอบ
[−14]. The second term is (−3) × 4 = −12, so the entry is −2 − 12 = −14.
การ์ด 38
คำถาม
Calculate AI and IA for these matrices.
A = [ 2 -1 ] [ 4 3 ] I = [ 1 0 ] [ 0 1 ]คำตอบ
Both equal A:
[ 2 -1 ] [ 4 3 ]Multiplication by this 2 × 2 identity matrix preserves A in either order.
การ์ด 39
คำถาม
Find entry (3, 1) of AB. Indices start at 1.
A = [ 2 0 ] [ -1 4 ] [ 5 -2 ] B = [ 3 ] [ 7 ]คำตอบ
1. Row 3 of A meets column 1 of B: 5 × 3 + (−2) × 7 = 15 − 14 = 1.
การ์ด 40
คำถาม
Calculate AB.
A = [ 3 -1 ] [ 2 4 ] B = [ 2 ] [ -3 ]คำตอบ
[ 9 ] [ -8 ]The rows give 6 + 3 = 9 and 4 − 12 = −8. The result has shape 2 × 1.
40 การ์ด
Matrix Multiplication Flashcards: Dimensions, Entries & Order
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