Matrix multiplication (*) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the * operator do in MATLAB when used between two matrices?

It performs matrix multiplication, combining rows of the first matrix with columns of the second matrix to produce a new matrix.

beginner

What condition must two matrices meet to be multiplied using * in MATLAB?

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

beginner

Given matrices A (2x3) and B (3x2), what will be the size of the result A * B?

The result will be a 2x2 matrix because the outer dimensions are taken from A's rows and B's columns.

beginner

What happens if you try to multiply two matrices with incompatible sizes in MATLAB?

MATLAB will give an error saying the inner matrix dimensions must agree.

intermediate

How is matrix multiplication different from element-wise multiplication in MATLAB?

Matrix multiplication (*) combines rows and columns following linear algebra rules, while element-wise multiplication (.*) multiplies corresponding elements directly.

What must be true for matrices A and B to multiply as A * B in MATLAB?

A Number of columns in A equals number of rows in B

B Number of rows in A equals number of rows in B

C Number of columns in A equals number of columns in B

D Matrices must be square

Matrix multiplication requires the inner dimensions to match: columns of A must equal rows of B.

If A is 4x2 and B is 2x3, what is the size of A * B?

A 4x2

B 2x2

C 2x3

D 4x3

The result size is rows of A by columns of B, so 4x3.

What operator does MATLAB use for element-wise multiplication?

A .*

B *

C +

D /

Element-wise multiplication uses .*, while * is for matrix multiplication.

What error occurs if you multiply incompatible matrices with * in MATLAB?

A Index exceeds matrix dimensions

B Inner matrix dimensions must agree

C Undefined function or variable

D Syntax error

MATLAB reports 'Inner matrix dimensions must agree' when sizes don't match for multiplication.

Which of these is true about matrix multiplication in MATLAB?

A It multiplies corresponding elements only

B It requires matrices to be square

C It combines rows of first with columns of second matrix

D It always produces a square matrix

Matrix multiplication combines rows of the first matrix with columns of the second.

Explain how matrix multiplication works in MATLAB and what conditions must be met for it to be valid.

Think about how rows and columns combine and what sizes must match.

Describe the difference between matrix multiplication (*) and element-wise multiplication (.*) in MATLAB.

Consider how the operations treat the matrices and their 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