Matrix rank and null space in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the rank of matrix A.

MATLAB

r = [1](A);

Drag options to blanks, or click blank then click option'

Arank

Bnull

Csize

Ddet

Attempts:

3 left

💡 Hint

Common Mistakes

Using null instead of rank.

Using size which returns matrix dimensions, not rank.

The rank function returns the rank of matrix A.

2fill in blank

medium

Complete the code to find the null space of matrix B.

MATLAB

N = [1](B);

Drag options to blanks, or click blank then click option'

Adet

Brank

Cinv

Dnull

Attempts:

3 left

💡 Hint

Common Mistakes

Using rank instead of null.

Using inv which requires a square invertible matrix.

The null function returns a basis for the null space of matrix B.

3fill in blank

hard

Fix the error in the code to compute the rank of matrix C.

MATLAB

r = rank[1]C[2];

Drag options to blanks, or click blank then click option'

A(

B)

C[

Attempts:

3 left

💡 Hint

Common Mistakes

Using square brackets or curly braces instead of parentheses.

Omitting parentheses entirely.

Function calls in MATLAB require parentheses around the input argument, like rank(C).

4fill in blank

hard

Fill both blanks to create a dictionary (struct) of ranks for matrices M and N.

MATLAB

ranks.M = [1](M);
ranks.N = [2](N);

Drag options to blanks, or click blank then click option'

Arank

Bnull

Csize

Ddet

Attempts:

3 left

💡 Hint

Common Mistakes

Using null for one or both blanks.

Using size or det which do not return rank.

Use rank to get the rank of both matrices M and N.

5fill in blank

hard

Fill all three blanks to compute rank, null space, and size of matrix P.

MATLAB

r = [1](P);
n = [2](P);
s = [3](P);

Drag options to blanks, or click blank then click option'

Arank

Bnull

Csize

Ddet

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up rank and null.

Using det instead of size.

Use rank for rank, null for null space, and size for dimensions of P.

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