Matrix creation in MATLAB - Time & Space Complexity

Time Complexity: Matrix creation

O(n^2)

Understanding Time Complexity

When we create a matrix in MATLAB, the computer needs to set up space and fill values. We want to know how the time it takes changes as the matrix size grows.

How does the work grow when we make bigger matrices?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

n = 1000;
A = zeros(n, n);
for i = 1:n
    for j = 1:n
        A(i,j) = i + j;
    end
end

This code creates an n-by-n matrix and fills each element with the sum of its row and column indices.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Filling each element of the matrix inside nested loops.
• How many times: The inner operation runs n x n times, once for each element.

How Execution Grows With Input

As the matrix size n grows, the number of operations grows by the square of n because we fill every element in a two-dimensional grid.

Input Size (n)	Approx. Operations
10	100
100	10,000
1000	1,000,000

Pattern observation: When n increases ten times, the work increases one hundred times.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to create and fill the matrix grows proportionally to the square of its size.

Common Mistake

[X] Wrong: "Filling the matrix takes time proportional to n, not n squared."

[OK] Correct: Because the matrix has n rows and n columns, you must fill n x n elements, so the work grows with n squared, not just n.

Interview Connect

Understanding how matrix creation time grows helps you explain efficiency in tasks like image processing or data tables, where big matrices are common. It shows you can think about how code scales with input size.

Self-Check

"What if we only filled the diagonal elements of the matrix? How would the time complexity change?"

Practice

1. Which of the following correctly creates a 2x3 matrix in MATLAB?

easy

A. [1 2; 3 4 5]

B. [1, 2; 3, 4; 5, 6]

C. [1 2 3; 4 5 6]

D. [1; 2; 3; 4]

Solution

1. Step 1: Understand matrix dimensions

   A 2x3 matrix has 2 rows and 3 columns.

2. Step 2: Check each option's shape

   [1 2 3; 4 5 6] has 2 rows separated by semicolon and 3 columns separated by spaces, so it is 2x3.

3. Final Answer:

   [1 2 3; 4 5 6] -> Option C

4. Quick Check:

   2 rows and 3 columns = [1 2 3; 4 5 6] [OK]

Hint: Rows separated by semicolons, columns by spaces or commas [OK]

Common Mistakes:

• Using commas instead of semicolons for rows
• Mismatched number of columns in rows
• Confusing rows and columns

2. Which MATLAB code correctly creates a 3x3 identity matrix?

easy

A. eye(3)

B. zeros(3,3)

C. ones(3)

D. diag(3)

Solution

1. Step 1: Recall identity matrix function

   In MATLAB, eye(n) creates an n-by-n identity matrix.

2. Step 2: Check options

   eye(3) uses eye(3), which is the correct function for a 3x3 identity matrix.

3. Final Answer:

   eye(3) -> Option A

4. Quick Check:

   Identity matrix = eye(3) [OK]

Hint: Use eye(n) for identity matrix of size n [OK]

Common Mistakes:

• Using zeros or ones instead of eye
• Using diag with incorrect vector size
• Confusing identity with zero matrix

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

A = [1 2; 3 4];
B = zeros(2,3);
C = [A B];
disp(C)

medium

A. [1 2; 3 4; 0 0 0]

B. [1 2 0 0 0; 3 4 0 0 0]

C. [1 2; 3 4; 0 0]

D. Error: Matrix dimensions must agree.

Solution

1. Step 1: Understand matrix sizes

   A is 2x2, B is 2x3 of zeros.

2. Step 2: Concatenate horizontally

   [A B] joins matrices side by side, resulting in 2 rows and 5 columns.

3. Final Answer:

   [1 2 0 0 0; 3 4 0 0 0] -> Option B

4. Quick Check:

   Horizontal concat of 2x2 and 2x3 = 2x5 matrix [OK]

Hint: Horizontal concat needs same row count [OK]

Common Mistakes:

• Trying to concat matrices with different row counts vertically
• Confusing zeros size
• Expecting error instead of valid concat

4. Identify the error in this MATLAB code:

M = [1 2 3; 4 5];

medium

A. Missing semicolon between rows

B. Matrix elements must be separated by commas

C. Use parentheses instead of square brackets

D. Rows have different number of columns

Solution

1. Step 1: Check matrix row sizes

   First row has 3 elements, second row has 2 elements.

2. Step 2: Understand MATLAB matrix rules

   All rows must have the same number of columns; here they differ.

3. Final Answer:

   Rows have different number of columns -> Option D

4. Quick Check:

   Matrix rows must match column count [OK]

Hint: All rows must have equal columns in matrix [OK]

Common Mistakes:

• Thinking commas are required between rows
• Using parentheses instead of brackets
• Ignoring unequal row lengths

5. You want to create a 4x4 matrix with 1s on the diagonal and 0.5 everywhere else. Which MATLAB code achieves this?

hard

A. M = 0.5*ones(4); M(eye(4) == 1) = 1;

B. M = eye(4) + 0.5;

C. M = 0.5*ones(4) + eye(4);

D. M = ones(4); M(eye(4) == 1) = 0.5;

Solution

1. Step 1: Create a 4x4 matrix of 0.5

   Using 0.5*ones(4) sets all elements to 0.5.

2. Step 2: Replace diagonal elements with 1

   Using logical indexing M(eye(4) == 1) = 1; sets diagonal to 1.

3. Final Answer:

   M = 0.5*ones(4); M(eye(4) == 1) = 1; -> Option A

4. Quick Check:

   Diagonal 1, others 0.5 = M = 0.5*ones(4); M(eye(4) == 1) = 1; [OK]

Hint: Set all to 0.5 then replace diagonal with 1 [OK]

Common Mistakes:

• Adding eye(4) instead of replacing diagonal
• Mixing addition and multiplication incorrectly
• Setting diagonal to 0.5 instead of 1