Matrix multiplication (*) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of this matrix multiplication?

Consider the following MATLAB code multiplying two matrices. What is the resulting matrix?

MATLAB

A = [1 2; 3 4];
B = [5 6; 7 8];
C = A * B;
disp(C);

A [12 16; 28 40]

B [19 22; 43 50]

C [5 12; 21 32]

D [17 20; 39 46]

💡 Hint

Remember matrix multiplication sums the products of rows of the first matrix with columns of the second.

Matrix multiplication multiplies rows of A by columns of B. For example, element (1,1) is 1*5 + 2*7 = 19.

❓ Predict Output

intermediate

What error does this matrix multiplication code produce?

What error will this MATLAB code produce when run?

MATLAB

A = [1 2 3; 4 5 6];
B = [7 8; 9 10];
C = A * B;

A Error: Undefined function or variable 'B'.

B No error, output is a 2x2 matrix.

C Error: Matrix dimensions must be square.

D Error: Inner matrix dimensions must agree.

💡 Hint

Check the number of columns in A and rows in B for multiplication compatibility.

Matrix multiplication requires the number of columns in A to equal the number of rows in B. Here, A is 2x3 and B is 2x2, so multiplication is invalid.

❓ Predict Output

advanced

What is the output of multiplying a matrix by identity matrix?

Given the code below, what is the output matrix C?

MATLAB

A = [3 4; 1 2];
I = eye(2);
C = A * I;
disp(C);

A [3 4; 1 2]

B [1 0; 0 1]

C [4 3; 2 1]

D [0 0; 0 0]

💡 Hint

Multiplying by the identity matrix should not change the original matrix.

The identity matrix acts like 1 in multiplication, so A * I = A.

❓ Predict Output

advanced

What is the output of this matrix multiplication with a transpose?

What is the output of the following MATLAB code?

MATLAB

A = [1 2 3];
B = [4; 5; 6];
C = A * B;
disp(C);

A 32

B [4 5 6; 8 10 12]

C [1 2 3; 4 5 6]

D Error: Inner matrix dimensions must agree.

💡 Hint

Check the dimensions of A and B before multiplication.

A is 1x3 and B is 3x1, so multiplication results in a 1x1 scalar: 1*4 + 2*5 + 3*6 = 32.

🧠 Conceptual

expert

How many elements does the resulting matrix have after multiplication?

If matrix A is 4x3 and matrix B is 3x5, how many elements will the matrix C = A * B have?

A 15

B 12

C 20

D 7

💡 Hint

The resulting matrix size is rows of A by columns of B.

Multiplying a 4x3 by a 3x5 matrix results in a 4x5 matrix, which has 4*5=20 elements.

Practice

1. Which of the following statements about matrix multiplication in MATLAB is correct?

easy

A. Matrix multiplication uses the operator .* in MATLAB.

B. The number of columns in the first matrix must equal the number of rows in the second matrix.

C. You can multiply any two matrices regardless of their sizes.

D. Matrix multiplication always produces a matrix of the same size as the first matrix.

Solution

1. Step 1: Understand matrix multiplication size rules

   Matrix multiplication requires the number of columns in the first matrix to match the number of rows in the second matrix.

2. Step 2: Identify correct operator and output size

   The operator for matrix multiplication is *, not .*, and the output size depends on the rows of the first and columns of the second matrix.

3. Final Answer:

   The number of columns in the first matrix must equal the number of rows in the second matrix. -> Option B

4. Quick Check:

   Matrix size rule = The number of columns in the first matrix must equal the number of rows in the second matrix. [OK]

Hint: Check matrix sizes: cols of first = rows of second [OK]

Common Mistakes:

• Confusing element-wise (.*) with matrix multiplication (*)
• Ignoring matrix size compatibility
• Assuming output size equals first matrix size

2. Which of the following MATLAB commands correctly multiplies matrix A by matrix B?

easy

A. C = A * B;

B. C = A .* B;

C. C = A + B;

D. C = A / B;

Solution

1. Step 1: Identify the correct operator for matrix multiplication

   In MATLAB, * is used for matrix multiplication, while .* is for element-wise multiplication.

2. Step 2: Confirm the correct syntax

   The command C = A * B; correctly multiplies matrices A and B.

3. Final Answer:

   C = A * B; -> Option A

4. Quick Check:

   Matrix multiply uses * operator [OK]

Hint: Use * for matrix multiply, .* for element-wise multiply [OK]

Common Mistakes:

• Using .* instead of * for matrix multiplication
• Using + or / operators incorrectly for multiplication
• Syntax errors with missing semicolons

3. What is the result of the following MATLAB code?

A = [1 2; 3 4];
B = [5 6; 7 8];
C = A * B;

medium

A. [[5 12]; [21 32]]

B. [[12 16]; [20 24]]

C. [[6 8]; [10 12]]

D. [[19 22]; [43 50]]

Solution

1. Step 1: Multiply rows of A by columns of B

   Calculate each element: C(1,1) = 1*5 + 2*7 = 19; C(1,2) = 1*6 + 2*8 = 22; C(2,1) = 3*5 + 4*7 = 43; C(2,2) = 3*6 + 4*8 = 50.

2. Step 2: Form the resulting matrix

   The resulting matrix C is [[19 22]; [43 50]].

3. Final Answer:

   [[19 22]; [43 50]] -> Option D

4. Quick Check:

   Matrix multiply result = [[19 22]; [43 50]] [OK]

Hint: Multiply rows by columns and sum products [OK]

Common Mistakes:

• Adding matrices instead of multiplying
• Using element-wise multiplication results
• Mixing up row and column indices

4. Consider the MATLAB code below. What is the error and how to fix it?

A = [1 2 3; 4 5 6];
B = [7 8; 9 10];
C = A * B;

medium

A. Error: Use .* instead of * for multiplication.

B. No error, code runs correctly.

C. Error: Matrix dimensions must agree. Fix by making B a 3xN matrix.

D. Error: Use plus (+) instead of multiplication.

Solution

1. Step 1: Check matrix sizes for multiplication

   A is 2x3, B is 2x2. Number of columns in A (3) does not equal number of rows in B (2), so multiplication is invalid.

2. Step 2: Fix the error by adjusting B's size

   To multiply, B must have 3 rows. For example, B could be 3x2 matrix.

3. Final Answer:

   Error: Matrix dimensions must agree. Fix by making B a 3xN matrix. -> Option C

4. Quick Check:

   Matrix size mismatch error = Error: Matrix dimensions must agree. Fix by making B a 3xN matrix. [OK]

Hint: Check cols of A = rows of B before multiplying [OK]

Common Mistakes:

• Ignoring dimension mismatch errors
• Trying to use .* instead of * to fix size errors
• Assuming multiplication works for any sizes

5. You have two matrices:

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

Calculate the product C = A * B and identify the element at position (2,1) in matrix C.

hard

A. 4

B. 5

C. 3

D. 6

Solution

1. Step 1: Verify matrix sizes for multiplication

   A is 2x3, B is 3x2, so multiplication is valid and result C will be 2x2.

2. Step 2: Calculate element C(2,1)

   Multiply row 2 of A by column 1 of B: (-1)*3 + 3*2 + 1*1 = -3 + 6 + 1 = 4.

3. Step 3: Recalculate carefully

   Recheck calculation: (-1)*3 = -3, 3*2 = 6, 1*1 = 1; sum = 4.

4. Final Answer:

   4 -> Option A

5. Quick Check:

   Sum products for C(2,1) = 4 [OK]

Hint: Multiply row 2 of A by column 1 of B and sum [OK]

Common Mistakes:

• Mixing row and column indices
• Forgetting negative signs in multiplication
• Adding wrong elements or missing terms