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