Matrix multiplication (*) in MATLAB

Introduction

Matrix multiplication lets you combine two matrices to get a new matrix that represents combined information.

When you want to combine data from two tables in a way that matches rows and columns mathematically.

When calculating transformations in graphics, like rotating or scaling shapes.

When solving systems of linear equations using matrices.

When working with neural networks to combine weights and inputs.

When performing operations in physics or engineering that involve vectors and matrices.

Syntax

MATLAB

C = A * B

A and B must have compatible sizes: number of columns in A must equal number of rows in B.

The result C will have the number of rows of A and the number of columns of B.

Examples

Multiply two 2x2 matrices to get a 2x2 matrix.

MATLAB

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

Multiply a 2x3 matrix by a 3x2 matrix to get a 2x2 matrix.

MATLAB

A = [1 2 3; 4 5 6];
B = [7 8; 9 10; 11 12];
C = A * B;

Sample Program

This program multiplies a 2x3 matrix A by a 3x2 matrix B and displays the result.

MATLAB

A = [1 0 2; -1 3 1];
B = [3 1; 2 1; 1 0];
C = A * B;
disp('Result of A * B is:');
disp(C);

Important Notes

If the sizes of A and B do not match for multiplication, MATLAB will give an error.

Use the element-wise multiplication operator (.*) if you want to multiply corresponding elements instead.

Matrix multiplication is not commutative: A * B is usually not the same as B * A.

Summary

Matrix multiplication combines two matrices into one by multiplying rows by columns.

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

Use * for matrix multiplication and .* for element-wise multiplication.

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