Matrix rank and null space in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of matrix rank calculation

What is the output of the following MATLAB code?

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
r = rank(A);
disp(r);

A 0

B 3

C 1

D 2

💡 Hint

Think about the linear dependence of rows in matrix A.

The matrix A has linearly dependent rows, so its rank is 2, not 3.

❓ Predict Output

intermediate

Null space dimension of a matrix

What is the size of the null space of matrix B?

MATLAB

B = [1 0 0; 0 1 0; 0 0 0];
N = null(B);
disp(size(N, 2));

A 2

B 0

C 1

D 3

💡 Hint

The null space dimension equals the number of free variables.

Matrix B has rank 2, so nullity = 3 - 2 = 1.

🔧 Debug

advanced

Identify the error in null space calculation

What error will this MATLAB code produce?

MATLAB

C = [1 2; 3 4];
N = null(C, 'r');
disp(N);

A Error: Undefined function or variable 'null'

B Error: Too many input arguments for null function

C No error, outputs null space basis

D Error: Matrix dimensions must agree

💡 Hint

Check the documentation for the null function's accepted arguments.

The null function in MATLAB does not accept a second argument 'r'.

🧠 Conceptual

advanced

Rank-nullity theorem application

Given a 4x5 matrix D with rank 3, what is the dimension of its null space?

A 2

B 3

C 1

D 4

💡 Hint

Use the formula: nullity = number_of_columns - rank.

Nullity = 5 - 3 = 2, so the null space dimension is 2.

❓ Predict Output

expert

Output of null space basis vectors

What is the output of this MATLAB code?

MATLAB

E = [1 2 3; 2 4 6; 3 6 9];
N = null(E);
disp(N);

A [ -0.8018; 0.2673; 0.5345 ]

B [ 0.2673; -0.8018; 0.5345 ]

C [ 0.5345; -0.8018; 0.2673 ]

D [ 0.5345; 0.2673; -0.8018 ]

💡 Hint

The null space basis vector is orthogonal to the rows of E.

The null function returns a basis vector approximately [-0.8018; 0.2673; 0.5345].

Practice

1. What does the rank function in MATLAB tell you about a matrix?

easy

A. The number of linearly independent rows or columns

B. The total number of elements in the matrix

C. The sum of all elements in the matrix

D. The determinant of the matrix

Solution

1. Step 1: Understand the definition of rank

   The rank of a matrix is the number of linearly independent rows or columns it has.

2. Step 2: Compare with other options

   Options A, B, and D describe different matrix properties, not rank.

3. Final Answer:

   The number of linearly independent rows or columns -> Option A

4. Quick Check:

   Rank = independent directions [OK]

Hint: Rank counts independent rows or columns [OK]

Common Mistakes:

• Confusing rank with matrix size
• Thinking rank is the sum of elements
• Mixing rank with determinant

2. Which MATLAB command correctly computes the null space of a matrix A?

easy

A. rank(A)

B. det(A)

C. null(A)

D. inv(A)

Solution

1. Step 1: Recall MATLAB functions for matrix properties

   null(A) returns the null space vectors of matrix A.

2. Step 2: Identify incorrect options

   rank(A) gives rank, det(A) gives determinant, inv(A) gives inverse if it exists.

3. Final Answer:

   null(A) -> Option C

4. Quick Check:

   null space = null(A) [OK]

Hint: Use null(A) to find null space vectors [OK]

Common Mistakes:

• Using rank(A) instead of null(A)
• Confusing determinant with null space
• Trying to invert a non-square matrix

3. Given the matrix A = [1 2 3; 2 4 6; 3 6 9], what is the output of rank(A) in MATLAB?

medium

A. 1

B. 2

C. 3

D. 0

Solution

1. Step 1: Analyze matrix linear dependence

   Rows 2 and 3 are multiples of row 1, so only 1 row is independent. But row 1 and 2 are linearly dependent, so rank is less than 3.

2. Step 2: Determine rank

   Actually, rows 2 and 3 are multiples of row 1, so rank is 1, but check carefully: row 2 = 2*row 1, row 3 = 3*row 1, so only one independent row.

3. Final Answer:

   2 -> Option B

4. Quick Check:

   Rank = 2 for dependent rows [OK]

Hint: Check if rows are multiples to find rank fast [OK]

Common Mistakes:

• Assuming rank equals number of rows
• Ignoring linear dependence
• Confusing rank with matrix size

4. Consider the MATLAB code:

A = [1 2 3; 4 5 6];
N = null(A);
disp(size(N));

What is the correct output of size(N)?

medium

A. [3 1]

B. [3 2]

C. [2 1]

D. [2 3]

Solution

1. Step 1: Determine matrix size and rank

   Matrix A is 2x3. Rank is 2 because rows are independent.

2. Step 2: Calculate null space dimension

   Null space dimension = number of columns - rank = 3 - 2 = 1. So null(A) returns a 3x1 matrix.

3. Final Answer:

   [3 1] -> Option A

4. Quick Check:

   Null space size = columns x (columns - rank) [OK]

Hint: Null space columns = cols - rank [OK]

Common Mistakes:

• Confusing rows and columns in size
• Forgetting null space dimension formula
• Expecting null space to be square

5. You have a matrix B of size 4x5 with rank(B) = 3. Which MATLAB command gives the dimension of the null space of B?

hard

A. size(B, 2) + rank(B)

B. rank(B)

C. size(B, 1) - rank(B)

D. size(null(B), 2)

Solution

1. Step 1: Understand null space dimension

   Null space dimension = number of columns - rank = 5 - 3 = 2.

2. Step 2: Use MATLAB to find null space dimension

   null(B) returns a matrix whose columns form a basis for the null space. The number of columns is the null space dimension. So size(null(B), 2) gives the dimension.

3. Final Answer:

   size(null(B), 2) -> Option D

4. Quick Check:

   Null space dimension = size(null(B), 2) [OK]

Hint: Count columns of null(B) for null space dimension [OK]

Common Mistakes:

• Using rank(B) instead of null space size
• Subtracting rows from rank
• Adding rank and columns incorrectly