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.
حول هذه الرزمة
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.
بطاقات هذه الرزمة
البطاقة ١
السؤال
What are the dimensions (rows × columns) of this matrix?
[ 6 -1 0 ] [ 2 3 5 ]الإجابة
2 × 3. There are two rows and three columns.
البطاقة ٢
السؤال
A is m × n and B is n × p. What are the dimensions of AB?
الإجابة
m × p. The shared inner dimension n makes the product defined; the outer dimensions give its shape. All dimensions here are positive integers.
البطاقة ٣
السؤال
For a defined matrix product C = AB, which parts of A and B determine entry C(i, j)? Indices start at 1.
الإجابة
Row i of A and column j of B. Multiply corresponding entries, then add every pairwise product.
البطاقة ٤
السؤال
A is 4 × 2 and B is 2 × 3. Is AB defined, and what is its shape?
الإجابة
Yes: 4 × 3. The inner dimensions are both 2.
البطاقة ٥
السؤال
Find entry (2, 1) of AB. Indices start at 1.
A = [ -1 4 ] [ 3 2 ] B = [ 5 -2 ] [ 1 6 ]الإجابة
17. Use row 2 of A and column 1 of B: 3 × 5 + 2 × 1 = 17.
البطاقة ٦
السؤال
A is 2 × 3 and B is 2 × 4. Is AB defined?
الإجابة
No. A has 3 columns, but B has 2 rows. The inner dimensions must match.
البطاقة ٧
السؤال
Calculate AB, keeping the result in matrix brackets.
A = [ 2 -1 3 ] B = [ 4 ] [ 5 ] [ -2 ]الإجابة
[ -3 ]The 1 × 1 entry is 2 × 4 + (−1) × 5 + 3 × (−2) = −3.
البطاقة ٨
السؤال
Calculate AB. Is the product undefined or a valid matrix?
A = [ 0 2 ] B = [ 3 ] [ 0 ]الإجابة
A valid 1 × 1 matrix:
[ 0 ]The dimensions match, and 0 × 3 + 2 × 0 = 0.
البطاقة ٩
السؤال
Find entry (1, 2) of AB. Indices start at 1.
A = [ 1 -2 4 ] [ 3 0 -1 ] B = [ 2 5 ] [ -1 3 ] [ 4 -2 ]الإجابة
−9. Row 1 meets column 2: 1 × 5 + (−2) × 3 + 4 × (−2) = 5 − 6 − 8 = −9.
البطاقة ١٠
السؤال
A is 3 × k and B is 5 × 2. AB is defined. What is k?
الإجابة
k = 5. A must have as many columns as B has rows. AB then has shape 3 × 2.
البطاقة ١١
السؤال
Calculate AB.
A = [ 3 ] [ -2 ] B = [ 1 4 ]الإجابة
[ 3 12 ] [ -2 -8 ]A 2 × 1 matrix times a 1 × 2 matrix gives a 2 × 2 matrix. Each entry has one pairwise product.
البطاقة ١٢
السؤال
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 ]الإجابة
13. Ordinary matrix multiplication uses row 1 against column 1: 2 × 4 + 1 × 5 = 13. Multiplying matching positions omits the second term.
البطاقة ١٣
السؤال
A is 2 × 3 and B is 3 × 2. Give the shape of AB and the shape of BA, or say undefined.
الإجابة
AB: 2 × 2. BA: 3 × 3. Both inner-dimension checks pass, but reversing the order changes the output shape.
البطاقة ١٤
السؤال
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 ]الإجابة
The missing term is 1 × (−5) = −5; the result is [−3]. All three pairs contribute: 8 − 6 − 5 = −3.
البطاقة ١٥
السؤال
Calculate AB.
A = [ 1 0 2 ] [ -1 3 1 ] B = [ 2 -1 ] [ 0 4 ] [ 3 2 ]الإجابة
[ 8 3 ] [ 1 15 ]The entries are 2 + 0 + 6, −1 + 0 + 4, −2 + 0 + 3, and 1 + 12 + 2.
البطاقة ١٦
السؤال
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 ]الإجابة
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.
البطاقة ١٧
السؤال
AB = [13]. Find x.
A = [ x 2 ] B = [ 3 ] [ -1 ]الإجابة
x = 5. The entry equation is 3x − 2 = 13. Substitution gives 3 × 5 − 2 = 13.
البطاقة ١٨
السؤال
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 ]الإجابة
−2. Entry (2, 1) needs row 2 of A: 4 × 1 + 3 × (−2) = −2.
البطاقة ١٩
السؤال
A is 2 × 3 and B is 4 × 2. Give the shape of AB and BA, or say undefined.
الإجابة
AB: undefined. BA: 4 × 3. For AB, 3 ≠ 4. For BA, the inner dimensions are both 2.
البطاقة ٢٠
السؤال
Calculate AB.
A = [ -2 1 ] [ 3 0 ] B = [ 4 -1 ] [ 2 5 ]الإجابة
[ -6 7 ] [ 12 -3 ]Row 1 gives −8 + 2 and 2 + 5; row 2 gives 12 + 0 and −3 + 0.
البطاقة ٢١
السؤال
A is m × 3, B is 3 × 4, and AB is 2 × 4. What is m?
الإجابة
m = 2. AB keeps the number of rows of the left matrix A.
البطاقة ٢٢
السؤال
Do AB and BA agree for this particular pair?
A = [ 2 0 ] [ 0 -1 ] B = [ 3 0 ] [ 0 4 ]الإجابة
Yes. Both products are:
[ 6 0 ] [ 0 -4 ]Some matrix pairs commute. Matrix multiplication is not commutative in general.
البطاقة ٢٣
السؤال
Find entry (3, 2) of AB. Indices start at 1.
A = [ 1 2 ] [ 0 -3 ] [ -2 4 ] B = [ 3 1 -1 ] [ 2 -2 5 ]الإجابة
−10. Use row 3 and column 2: (−2) × 1 + 4 × (−2) = −10.
البطاقة ٢٤
السؤال
A is 2 × 3 and B is 4 × 5. Is either AB or BA defined?
الإجابة
Neither is defined. AB fails because 3 ≠ 4; BA fails because 5 ≠ 2.
البطاقة ٢٥
السؤال
Entry (2, 1) of AB is 22. Find x.
A = [ 2 -1 ] [ 1 3 ] B = [ x 4 ] [ 5 -2 ]الإجابة
x = 7. Entry (2, 1) gives x + 3 × 5 = 22. Substitution gives 7 + 15 = 22.
البطاقة ٢٦
السؤال
Calculate both AB and BA.
A = [ 1 -2 ] B = [ 3 ] [ 4 ]الإجابة
AB:
[ -5 ]BA:
[ 3 -6 ] [ 4 -8 ]AB sums 3 − 8. BA is a 2 × 2 outer product.
البطاقة ٢٧
السؤال
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 ]الإجابة
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.
البطاقة ٢٨
السؤال
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.
الإجابة
3 × 5. The matching inner dimensions allow multiplication; the outer dimensions determine the result.
البطاقة ٢٩
السؤال
Calculate AB.
A = [ 2 0 -1 ] B = [ 3 1 ] [ 4 -2 ] [ -2 5 ]الإجابة
[ 8 -3 ]The entries are 6 + 0 + 2 = 8 and 2 + 0 − 5 = −3. The result has shape 1 × 2.
البطاقة ٣٠
السؤال
A is 2 × 4 and B is 4 × 3. How many pairwise products must be added to calculate entry (2, 3) of AB?
الإجابة
Four. Row 2 of A and column 3 of B each contain four entries, so every output entry sums four pairwise products.
البطاقة ٣١
السؤال
AB = [10]. Find x.
A = [ 2 x -1 ] B = [ 3 ] [ -2 ] [ 4 ]الإجابة
x = −4. The equation is 6 − 2x − 4 = 10, so −2x = 8. Substitution gives 6 + 8 − 4 = 10.
البطاقة ٣٢
السؤال
What is column 2 of AB?
A = [ 1 -2 4 ] [ 0 3 5 ] B = [ 2 0 ] [ -1 0 ] [ 6 0 ]الإجابة
[ 0 ] [ 0 ]Each entry in column 2 uses the all-zero column 2 of B, so every term is zero.
البطاقة ٣٣
السؤال
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 ]الإجابة
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.
البطاقة ٣٤
السؤال
A and B are both 2 × 3. Does having the same shape make AB defined?
الإجابة
No. The inner dimensions are 3 and 2, so AB is undefined. Equal shapes alone do not ensure compatibility.
البطاقة ٣٥
السؤال
Find entry (1, 2) of AB. Indices start at 1.
A = [ 2 1 ] [ -1 3 ] B = [ 4 -2 ] [ 2 4 ]الإجابة
0. The entry is 2 × (−2) + 1 × 4 = −4 + 4 = 0. Cancellation is a valid result.
البطاقة ٣٦
السؤال
A is 4 × 2 and B is k × 3. AB is defined and has shape 4 × 3. What is k?
الإجابة
k = 2. B must have two rows to match A’s two columns.
البطاقة ٣٧
السؤال
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 ]الإجابة
[−14]. The second term is (−3) × 4 = −12, so the entry is −2 − 12 = −14.
البطاقة ٣٨
السؤال
Calculate AI and IA for these matrices.
A = [ 2 -1 ] [ 4 3 ] I = [ 1 0 ] [ 0 1 ]الإجابة
Both equal A:
[ 2 -1 ] [ 4 3 ]Multiplication by this 2 × 2 identity matrix preserves A in either order.
البطاقة ٣٩
السؤال
Find entry (3, 1) of AB. Indices start at 1.
A = [ 2 0 ] [ -1 4 ] [ 5 -2 ] B = [ 3 ] [ 7 ]الإجابة
1. Row 3 of A meets column 1 of B: 5 × 3 + (−2) × 7 = 15 − 14 = 1.
البطاقة ٤٠
السؤال
Calculate AB.
A = [ 3 -1 ] [ 2 4 ] B = [ 2 ] [ -3 ]الإجابة
[ 9 ] [ -8 ]The rows give 6 + 3 = 9 and 4 − 12 = −8. The result has shape 2 × 1.
٤٠ بطاقة
Matrix Multiplication Flashcards: Dimensions, Entries & Order
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