Matrix Multiplication Flashcards: Dimensions, Entries & Order

Question 1

What are the dimensions (rows × columns) of this matrix?

[ 6  -1   0 ]
[ 2   3   5 ]

Accepted Answer

2 × 3. There are two rows and three columns.

Question 2

A is m × n and B is n × p. What are the dimensions of AB?

Accepted Answer

m × p. The shared inner dimension n makes the product defined; the outer dimensions give its shape. All dimensions here are positive integers.

Question 3

For a defined matrix product C = AB, which parts of A and B determine entry C(i, j)? Indices start at 1.

Accepted Answer

Row i of A and column j of B. Multiply corresponding entries, then add every pairwise product.

Question 4

A is 4 × 2 and B is 2 × 3. Is AB defined, and what is its shape?

Accepted Answer

Yes: 4 × 3. The inner dimensions are both 2.

Question 5

Find entry (2, 1) of AB. Indices start at 1.

A = [ -1   4 ]
    [  3   2 ]

B = [ 5  -2 ]
    [ 1   6 ]

Accepted Answer

17. Use row 2 of A and column 1 of B: 3 × 5 + 2 × 1 = 17.

Question 6

A is 2 × 3 and B is 2 × 4. Is AB defined?

Accepted Answer

No. A has 3 columns, but B has 2 rows. The inner dimensions must match.

Question 7

Calculate AB, keeping the result in matrix brackets.

A = [ 2  -1   3 ]

B = [  4 ]
    [  5 ]
    [ -2 ]

Accepted Answer

[ -3 ]

The 1 × 1 entry is 2 × 4 + (−1) × 5 + 3 × (−2) = −3.

Question 8

Calculate AB. Is the product undefined or a valid matrix?

A = [ 0   2 ]

B = [ 3 ]
    [ 0 ]

Accepted Answer

A valid 1 × 1 matrix:

[ 0 ]

The dimensions match, and 0 × 3 + 2 × 0 = 0.

Question 9

Find entry (1, 2) of AB. Indices start at 1.

A = [ 1  -2   4 ]
    [ 3   0  -1 ]

B = [  2   5 ]
    [ -1   3 ]
    [  4  -2 ]

Accepted Answer

−9. Row 1 meets column 2: 1 × 5 + (−2) × 3 + 4 × (−2) = 5 − 6 − 8 = −9.

Question 10

A is 3 × k and B is 5 × 2. AB is defined. What is k?

Accepted Answer

k = 5. A must have as many columns as B has rows. AB then has shape 3 × 2.

Question 11

Calculate AB.

A = [  3 ]
    [ -2 ]

B = [ 1   4 ]

Accepted Answer

[  3   12 ]
[ -2   -8 ]

A 2 × 1 matrix times a 1 × 2 matrix gives a 2 × 2 matrix. Each entry has one pairwise product.

Question 12

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 ]

Accepted Answer

13. Ordinary matrix multiplication uses row 1 against column 1: 2 × 4 + 1 × 5 = 13. Multiplying matching positions omits the second term.

Question 13

A is 2 × 3 and B is 3 × 2. Give the shape of AB and the shape of BA, or say undefined.

Accepted Answer

AB: 2 × 2. BA: 3 × 3. Both inner-dimension checks pass, but reversing the order changes the output shape.

Question 14

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 ]

Accepted Answer

The missing term is 1 × (−5) = −5; the result is [−3]. All three pairs contribute: 8 − 6 − 5 = −3.

Question 15

Calculate AB.

A = [  1   0   2 ]
    [ -1   3   1 ]

B = [ 2  -1 ]
    [ 0   4 ]
    [ 3   2 ]

Accepted Answer

[ 8    3 ]
[ 1   15 ]

The entries are 2 + 0 + 6, −1 + 0 + 4, −2 + 0 + 3, and 1 + 12 + 2.

Question 16

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 ]

Accepted Answer

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.

Question 17

AB = [13]. Find x.

A = [ x   2 ]

B = [  3 ]
    [ -1 ]

Accepted Answer

x = 5. The entry equation is 3x − 2 = 13. Substitution gives 3 × 5 − 2 = 13.

Question 18

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 ]

Accepted Answer

−2. Entry (2, 1) needs row 2 of A: 4 × 1 + 3 × (−2) = −2.

Question 19

A is 2 × 3 and B is 4 × 2. Give the shape of AB and BA, or say undefined.

Accepted Answer

AB: undefined. BA: 4 × 3. For AB, 3 ≠ 4. For BA, the inner dimensions are both 2.

Question 20

Calculate AB.

A = [ -2   1 ]
    [  3   0 ]

B = [ 4  -1 ]
    [ 2   5 ]

Accepted Answer

[ -6    7 ]
[ 12   -3 ]

Row 1 gives −8 + 2 and 2 + 5; row 2 gives 12 + 0 and −3 + 0.

Question 21

A is m × 3, B is 3 × 4, and AB is 2 × 4. What is m?

Accepted Answer

m = 2. AB keeps the number of rows of the left matrix A.

Question 22

Do AB and BA agree for this particular pair?

A = [ 2   0 ]
    [ 0  -1 ]

B = [ 3   0 ]
    [ 0   4 ]

Accepted Answer

Yes. Both products are:

[ 6    0 ]
[ 0   -4 ]

Some matrix pairs commute. Matrix multiplication is not commutative in general.

Question 23

Find entry (3, 2) of AB. Indices start at 1.

A = [  1   2 ]
    [  0  -3 ]
    [ -2   4 ]

B = [ 3   1  -1 ]
    [ 2  -2   5 ]

Accepted Answer

−10. Use row 3 and column 2: (−2) × 1 + 4 × (−2) = −10.

Question 24

A is 2 × 3 and B is 4 × 5. Is either AB or BA defined?

Accepted Answer

Neither is defined. AB fails because 3 ≠ 4; BA fails because 5 ≠ 2.

Question 25

Entry (2, 1) of AB is 22. Find x.

A = [ 2  -1 ]
    [ 1   3 ]

B = [ x   4 ]
    [ 5  -2 ]

Accepted Answer

x = 7. Entry (2, 1) gives x + 3 × 5 = 22. Substitution gives 7 + 15 = 22.

Question 26

Calculate both AB and BA.

A = [ 1  -2 ]

B = [ 3 ]
    [ 4 ]

Accepted Answer

AB:

[ -5 ]

BA:

[ 3   -6 ]
[ 4   -8 ]

AB sums 3 − 8. BA is a 2 × 2 outer product.

Question 27

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 ]

Accepted Answer

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.

Question 28

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.

Accepted Answer

3 × 5. The matching inner dimensions allow multiplication; the outer dimensions determine the result.

Question 29

Calculate AB.

A = [ 2   0  -1 ]

B = [  3   1 ]
    [  4  -2 ]
    [ -2   5 ]

Accepted Answer

[ 8   -3 ]

The entries are 6 + 0 + 2 = 8 and 2 + 0 − 5 = −3. The result has shape 1 × 2.

Question 30

A is 2 × 4 and B is 4 × 3. How many pairwise products must be added to calculate entry (2, 3) of AB?

Accepted Answer

Four. Row 2 of A and column 3 of B each contain four entries, so every output entry sums four pairwise products.

Question 31

AB = [10]. Find x.

A = [ 2   x  -1 ]

B = [  3 ]
    [ -2 ]
    [  4 ]

Accepted Answer

x = −4. The equation is 6 − 2x − 4 = 10, so −2x = 8. Substitution gives 6 + 8 − 4 = 10.

Question 32

What is column 2 of AB?

A = [ 1  -2   4 ]
    [ 0   3   5 ]

B = [  2   0 ]
    [ -1   0 ]
    [  6   0 ]

Accepted Answer

[ 0 ]
[ 0 ]

Each entry in column 2 uses the all-zero column 2 of B, so every term is zero.

Question 33

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 ]

Accepted Answer

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.

Question 34

A and B are both 2 × 3. Does having the same shape make AB defined?

Accepted Answer

No. The inner dimensions are 3 and 2, so AB is undefined. Equal shapes alone do not ensure compatibility.

Question 35

Find entry (1, 2) of AB. Indices start at 1.

A = [  2   1 ]
    [ -1   3 ]

B = [ 4  -2 ]
    [ 2   4 ]

Accepted Answer

0. The entry is 2 × (−2) + 1 × 4 = −4 + 4 = 0. Cancellation is a valid result.

Question 36

A is 4 × 2 and B is k × 3. AB is defined and has shape 4 × 3. What is k?

Accepted Answer

k = 2. B must have two rows to match A’s two columns.

Question 37

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 ]

Accepted Answer

[−14]. The second term is (−3) × 4 = −12, so the entry is −2 − 12 = −14.

Question 38

Calculate AI and IA for these matrices.

A = [ 2  -1 ]
    [ 4   3 ]

I = [ 1   0 ]
    [ 0   1 ]

Accepted Answer

Both equal A:

[ 2  -1 ]
[ 4   3 ]

Multiplication by this 2 × 2 identity matrix preserves A in either order.

Question 39

Find entry (3, 1) of AB. Indices start at 1.

A = [  2   0 ]
    [ -1   4 ]
    [  5  -2 ]

B = [ 3 ]
    [ 7 ]

Accepted Answer

1. Row 3 of A meets column 1 of B: 5 × 3 + (−2) × 7 = 15 − 14 = 1.

Question 40

Calculate AB.

A = [ 3  -1 ]
    [ 2   4 ]

B = [  2 ]
    [ -3 ]

Accepted Answer

[  9 ]
[ -8 ]

The rows give 6 + 3 = 9 and 4 − 12 = −8. The result has shape 2 × 1.

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Matrix Multiplication Flashcards: Dimensions, Entries & Order