Matrix transpose in MATLAB - Step-by-Step Execution

Concept Flow - Matrix transpose

Start with matrix A

↓

Read element A(i,j)

↓

Place element at position B(j,i)

↓

Repeat for all i,j

↓

Result: matrix B is transpose of A

The process reads each element of the original matrix and places it in the flipped position to create the transpose.

Execution Sample

MATLAB

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

This code creates a 2x3 matrix A, transposes it to a 3x2 matrix B, and displays B.

Execution Table

Step	Action	Element Read A(i,j)	Element Placed B(j,i)	Matrix B State
1	Read A(1,1)	1	B(1,1) = 1	[1 0; 0 0; 0 0]
2	Read A(1,2)	2	B(2,1) = 2	[1 0; 2 0; 0 0]
3	Read A(1,3)	3	B(3,1) = 3	[1 0; 2 0; 3 0]
4	Read A(2,1)	4	B(1,2) = 4	[1 4; 2 0; 3 0]
5	Read A(2,2)	5	B(2,2) = 5	[1 4; 2 5; 3 0]
6	Read A(2,3)	6	B(3,2) = 6	[1 4; 2 5; 3 6]
7	Transpose complete	-	-	Final B = [1 4; 2 5; 3 6]

💡 All elements of A read and placed in B with swapped indices.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	Final
A	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]	[1 2 3; 4 5 6]
B	empty 3x2	[1 0; 0 0; 0 0]	[1 0; 2 0; 0 0]	[1 0; 2 0; 3 0]	[1 4; 2 0; 3 0]	[1 4; 2 5; 3 0]	[1 4; 2 5; 3 6]	[1 4; 2 5; 3 6]

Key Moments - 3 Insights

Why does the size of matrix B change compared to A?

Because transpose swaps rows and columns, so a 2x3 matrix A becomes a 3x2 matrix B, as shown in the execution_table rows where indices are swapped.

Why is element A(1,2) placed at B(2,1) and not B(1,2)?

Transpose swaps the row and column indices, so element at row 1, column 2 in A goes to row 2, column 1 in B, as seen in step 2 of the execution_table.

Is the original matrix A changed after transpose?

No, A remains the same throughout, only B is created and filled, as shown in variable_tracker where A stays constant.

Visual Quiz - 3 Questions

Look at the execution_table at step 4, what is the value placed in B and at which position?

A B(2,1) = 4

B B(1,2) = 4

C B(2,1) = 2

D B(1,1) = 4

💡 Hint

Check step 4 row in execution_table under 'Element Placed B(j,i)' column.

At which step does the last element of A get placed into B?

A Step 6

B Step 5

C Step 3

D Step 7

💡 Hint

Look at the last element read and placed in execution_table rows.

If matrix A was 3x2 instead of 2x3, what would be the size of B after transpose?

A 3x2

B 2x2

C 2x3

D 3x3

💡 Hint

Transpose swaps rows and columns, so check variable_tracker for size changes.

Concept Snapshot

Matrix transpose swaps rows and columns.
Syntax in MATLAB: B = A';
If A is m x n, B becomes n x m.
Each element A(i,j) moves to B(j,i).
Original matrix A stays unchanged.
Transpose is useful for flipping matrix orientation.

Full Transcript

Matrix transpose means flipping a matrix over its diagonal. We start with matrix A. For each element at row i and column j in A, we place it at row j and column i in matrix B. This swaps rows and columns. For example, a 2 by 3 matrix becomes a 3 by 2 matrix after transpose. The original matrix A does not change. The code A = [1 2 3; 4 5 6]; B = A'; disp(B); creates matrix A, transposes it to B, and displays B. Step by step, each element of A is read and placed in B at swapped indices. After all elements are processed, B is the transpose of A.

Practice

1. What does the transpose operation do to a matrix in MATLAB?

easy

A. It swaps the rows and columns of the matrix.

B. It multiplies all elements by -1.

C. It adds a new row to the matrix.

D. It deletes the last column of the matrix.

Solution

1. Step 1: Understand matrix structure

   A matrix has rows and columns arranged in a grid.

2. Step 2: Define transpose operation

   Transposing swaps each row with the corresponding column, flipping the matrix over its diagonal.

3. Final Answer:

   It swaps the rows and columns of the matrix. -> Option A

4. Quick Check:

   Transpose = swap rows and columns [OK]

Hint: Transpose flips rows and columns in a matrix [OK]

Common Mistakes:

• Thinking transpose multiplies elements
• Confusing transpose with adding rows
• Assuming transpose deletes columns

2. Which of the following is the correct MATLAB syntax to transpose a matrix A?

easy

A. A.T

B. trans(A)

C. A'

D. A^T

Solution

1. Step 1: Recall MATLAB transpose operators

   In MATLAB, the apostrophe ' is used to transpose a matrix.

2. Step 2: Common incorrect syntax

   A.T is Python syntax; trans(A) and A^T are invalid in MATLAB.

3. Final Answer:

   A' -> Option C

4. Quick Check:

   Use apostrophe (') for transpose [OK]

Hint: Use apostrophe (') to transpose matrix in MATLAB [OK]

Common Mistakes:

• Using A^T which is invalid syntax
• Using A.T (Python syntax)
• Using trans(A) which is undefined

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

A = [1 2; 3 4];
B = A';
disp(B);

medium

A. [1 3; 2 4]

B. [1 2; 3 4]

C. [1 3; 4 2]

D. [2 1; 4 3]

Solution

1. Step 1: Understand matrix A

   Matrix A is 2x2: first row [1 2], second row [3 4].

2. Step 2: Apply transpose operation

   Transposing swaps rows and columns: first column becomes first row, second column becomes second row. So B = [1 3; 2 4].

3. Final Answer:

   [1 3; 2 4] -> Option A

4. Quick Check:

   Transpose swaps rows and columns [OK]

Hint: Swap rows and columns to find transpose [OK]

Common Mistakes:

• Mixing rows and columns incorrectly
• Confusing transpose with element-wise operations
• Reading matrix elements in wrong order

4. Identify the error in this MATLAB code snippet:

A = [1 2 3; 4 5 6];
B = A';
C = B(1,3);

medium

A. No error; code runs correctly.

B. Indexing error: B(1,3) is out of bounds after transpose.

C. Matrix A is not defined properly.

D. Syntax error: transpose operator used incorrectly.

Solution

1. Step 1: Analyze matrix A size

   A is 2 rows by 3 columns.

2. Step 2: Transpose matrix A

   After transpose, B is 3 rows by 2 columns.

3. Step 3: Check indexing B(1,3)

   Since B has only 2 columns, B(1,3) is out of bounds and will cause an error.

4. Final Answer:

   Indexing error: B(1,3) is out of bounds after transpose. -> Option B

5. Quick Check:

   Check matrix size before indexing [OK]

Hint: Check matrix size after transpose before indexing [OK]

Common Mistakes:

• Assuming size stays same after transpose
• Ignoring column count when indexing
• Confusing row and column indices

5. Given a complex matrix C = [1+2i, 3-4i; 5+6i, 7-8i], which MATLAB expression returns the transpose without conjugation?

hard

A. C'

B. C.T

C. ctranspose(C)

D. C.'

Solution

1. Step 1: Understand transpose with complex numbers

   In MATLAB, C' transposes and conjugates complex matrices.

2. Step 2: Identify transpose without conjugation

   The operator .' transposes without conjugating complex elements.

3. Step 3: Check other options

   ctranspose(C) performs conjugate transpose; C.T is invalid syntax.

4. Final Answer:

   C.' -> Option D

5. Quick Check:

   Use dot-apostrophe (.' ) for transpose without conjugation [OK]

Hint: Use dot-apostrophe (.' ) to transpose without conjugation [OK]

Common Mistakes:

• Using apostrophe (') which conjugates
• Confusing transpose and conjugate
• Using ctranspose(C) which conjugates