Matrix concatenation in MATLAB - Step-by-Step Execution

Concept Flow - Matrix concatenation

Define matrices A and B

↓

Choose concatenation direction

↓

Horizontal concat

↓

Combine side-by-side

↓

Resulting matrix C

↓

Use C

Start with two matrices, decide to join them side-by-side (horizontal) or top-to-bottom (vertical), then combine to get a new matrix.

Execution Sample

MATLAB

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

Create two 2x2 matrices A and B, then concatenate horizontally into C and vertically into D.

Execution Table

Step	Operation	Matrices Involved	Resulting Matrix	Size
1	Define A	None	[1 2; 3 4]	2x2
2	Define B	None	[5 6; 7 8]	2x2
3	Horizontal concat C = [A B]	A, B	[1 2 5 6; 3 4 7 8]	2x4
4	Vertical concat D = [A; B]	A, B	[1 2; 3 4; 5 6; 7 8]	4x2
5	End	None	No more operations	-

💡 All concatenations done; matrices combined horizontally and vertically.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	Final
A	undefined	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]
B	undefined	undefined	[5 6; 7 8]	[5 6; 7 8]	[5 6; 7 8]	[5 6; 7 8]
C	undefined	undefined	undefined	[1 2 5 6; 3 4 7 8]	[1 2 5 6; 3 4 7 8]	[1 2 5 6; 3 4 7 8]
D	undefined	undefined	undefined	undefined	[1 2; 3 4; 5 6; 7 8]	[1 2; 3 4; 5 6; 7 8]

Key Moments - 2 Insights

Why does horizontal concatenation require the same number of rows in A and B?

Because when joining side-by-side, each row from A must line up with a row from B. If rows differ, MATLAB cannot align them, causing an error. See step 3 in execution_table where both A and B have 2 rows.

Why must vertical concatenation have the same number of columns?

When stacking matrices top-to-bottom, columns must match so each column aligns vertically. Step 4 shows A and B both have 2 columns, allowing vertical concatenation.

Visual Quiz - 3 Questions

Look at the execution_table at step 3. What is the size of matrix C after horizontal concatenation?

A 4x2

B 2x4

C 2x2

D 4x4

💡 Hint

Check the 'Size' column in execution_table row for step 3.

At which step does matrix D get its value?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for vertical concatenation operation in execution_table.

If matrix B had 3 rows instead of 2, what would happen at step 3?

A Error due to row mismatch

B Horizontal concatenation would succeed with larger C

C Vertical concatenation would fail instead

D No change, concatenation works

💡 Hint

Recall key_moments about row matching for horizontal concatenation.

Concept Snapshot

Matrix concatenation in MATLAB:
- Use [A B] for horizontal (side-by-side) concatenation.
- Use [A; B] for vertical (top-to-bottom) concatenation.
- Horizontal concat requires same number of rows.
- Vertical concat requires same number of columns.
- Resulting matrix size depends on concat direction.

Full Transcript

This visual trace shows how MATLAB concatenates matrices. First, matrices A and B are defined as 2x2. Then, horizontal concatenation [A B] joins them side-by-side, resulting in a 2x4 matrix C. Vertical concatenation [A; B] stacks them top-to-bottom, creating a 4x2 matrix D. Key rules are that horizontal concatenation needs matching rows, and vertical concatenation needs matching columns. The variable tracker shows how A, B, C, and D change step-by-step. The quiz checks understanding of sizes and errors if dimensions don't match.

Practice

1. Which of the following correctly describes horizontal concatenation of matrices in MATLAB?

easy

A. Using semicolons between matrices with the same number of columns

B. Using parentheses around matrices to join them

C. Using plus signs between matrices of any size

D. Using commas or spaces between matrices with the same number of rows

Solution

1. Step 1: Understand horizontal concatenation syntax

   In MATLAB, horizontal concatenation joins matrices side by side using commas or spaces.

2. Step 2: Check dimension requirements

   Matrices must have the same number of rows to concatenate horizontally.

3. Final Answer:

   Using commas or spaces between matrices with the same number of rows -> Option D

4. Quick Check:

   Horizontal concat = commas/spaces + same rows [OK]

Hint: Use commas or spaces for horizontal join, rows must match [OK]

Common Mistakes:

• Using semicolons for horizontal concatenation
• Joining matrices with different row counts
• Using plus sign instead of concatenation

2. Which of the following MATLAB commands correctly concatenates matrices A and B vertically?

easy

A. C = (A; B);

B. C = [A, B];

C. C = [A; B];

D. C = [A + B];

Solution

1. Step 1: Identify vertical concatenation syntax

   Vertical concatenation in MATLAB uses semicolons between matrices inside square brackets.

2. Step 2: Check the options

   C = [A; B]; uses [A; B], which is the correct syntax for vertical concatenation.

3. Final Answer:

   C = [A; B]; -> Option C

4. Quick Check:

   Vertical concat = semicolon inside brackets [OK]

Hint: Use semicolon inside brackets for vertical concatenation [OK]

Common Mistakes:

• Using commas for vertical concatenation
• Using parentheses instead of brackets
• Trying to add matrices with plus sign

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. [1 2 5 6; 3 4 7 8]

B. [1 2; 3 4; 5 6; 7 8]

C. [6 8; 10 12]

D. Error due to dimension mismatch

Solution

1. Step 1: Understand matrices A and B

   A and B are both 2x2 matrices.

2. Step 2: Apply horizontal concatenation

   Using [A, B] joins them side by side, resulting in a 2x4 matrix: first two columns from A, next two from B.

3. Final Answer:

   [1 2 5 6; 3 4 7 8] -> Option A

4. Quick Check:

   Horizontal concat of 2x2 + 2x2 = 2x4 matrix [OK]

Hint: Horizontal concat adds columns, rows stay same [OK]

Common Mistakes:

• Confusing horizontal with vertical concatenation
• Expecting sum of matrices instead of concatenation
• Thinking dimensions mismatch causes error here

4. Consider the following MATLAB code:

A = [1 2 3];
B = [4 5];
C = [A; B];

What is the issue with this code?

medium

A. Dimension mismatch error during vertical concatenation

B. No error, concatenation works fine

C. Syntax error due to missing commas

D. Runtime error due to invalid matrix elements

Solution

1. Step 1: Check sizes of A and B

   A is 1x3, B is 1x2, so they have different numbers of columns.

2. Step 2: Understand vertical concatenation requirements

   Vertical concatenation [A; B] requires matrices to have the same number of columns, which is not true here.

3. Final Answer:

   Dimension mismatch error during vertical concatenation -> Option A

4. Quick Check:

   Vertical concat needs equal columns [OK]

Hint: Vertical concat needs same columns, else error [OK]

Common Mistakes:

• Assuming MATLAB auto-fills missing elements
• Using commas instead of semicolons for vertical concat
• Ignoring dimension mismatch errors

5. You have two matrices:

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

Which MATLAB command correctly concatenates A and B horizontally without error?

hard

A. C = [A, B];

B. C = [A, B(:,1:2)];

C. C = [A; B];

D. C = [A; B(:,1:2)];

Solution

1. Step 1: Check dimensions of A and B

   A is 2x2, B is 2x3. Both have 2 rows.

2. Step 2: Horizontal concatenation requirements

   Horizontal concatenation requires same number of rows, which matches. However, A is 2x2 and B is 2x3, so concatenating [A, B] results in a 2x5 matrix, which is valid.

3. Step 3: Check other options

   B: Vertical concat, columns mismatch (2 vs 3), error. C: Concatenates A with first two columns of B, resulting in 2x4 matrix, also valid. D: Vertical concat, columns mismatch, error.

4. Step 4: Clarify the question's intent

   The question asks for concatenation without error. Both A and C produce valid concatenations. However, C = [A, B]; concatenates full B (2x3) with A (2x2), which is valid. C = [A, B(:,1:2)]; concatenates A with a sliced B (2x2), also valid.

5. Step 5: Resolve ambiguity

   Since both A and C are valid, but the question expects one correct answer, the safer choice is C, which explicitly slices B to match A's columns, ensuring no dimension mismatch in other contexts.

6. Final Answer:

   C = [A, B(:,1:2)]; -> Option B

7. Quick Check:

   Horizontal concat needs same rows; slicing columns ensures compatibility [OK]

Hint: Horizontal: same rows, columns can differ; slicing ensures compatibility [OK]

Common Mistakes:

• Trying to concatenate matrices with mismatched dimensions
• Ignoring dimension requirements for vertical concat
• Assuming MATLAB auto-adjusts matrix sizes