Rank, Invertibility and Linear Systems Practice Questions for GATE DA: 20+ Solved Questions with Step-by-Step Solutions

Solve 20+ Rank, Invertibility and Linear Systems practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Rank, Invertibility and Linear Systems

Chapter Roadmap

Rank, Invertibility and Linear Systems

Rank, Nullity and Matrix Polynomials
Solution spaces, Rank-Nullity Theorem, and matrix powers.

Invertibility and Rank-One Updates
Outer products

x x^{T}

, invertibility shifts, and quick eigenvalue tricks.

Goal: Move from calculating determinants to understanding the structure of linear transformations.

Rank, Nullity and the Solution Space

For any $m \times n$ matrix $A$ , the fundamental accounting rule of linear algebra is:

Rank (A) + Nullity (A) = n

Rank: Dimension of the column space (independent columns).

Nullity: Dimension of the null space (free variables in

A x = 0

$n$ : Total number of columns (variables).

Exam Insight: If

Rank (A) < n

, the system

A x = 0

has non-trivial (infinite) solutions.

Rank, Invertibility and Linear Systems: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Linear Algebra MCQ

For an $m \times n$ matrix $A$ , if the rank of $A$ is strictly less than $n$ , then the homogeneous system of linear equations $A x = 0$ has:

A.
No solution
B.
Exactly one solution
C.
Exactly $n$ solutions
D.
Infinitely many solutions

Correct Answer:

Step-by-Step Solution

Key idea: This question tests the fundamental implication of the Rank-Nullity Theorem on linear systems.

Step 1: For an $m \times n$ matrix $A$ , the Rank-Nullity Theorem states that $Rank (A) + Nullity (A) = n$ .

Step 2: We are given that $Rank (A) < n$ .

Step 3: This implies that $Nullity (A) = n - Rank (A) > 0$ .

Step 4: A positive nullity means the null space contains non-zero vectors, so the homogeneous system $A x = 0$ has non-trivial (infinitely many) solutions.

Answer: Infinitely many solutions

Question 2 · Linear Algebra MCQ

Let $A$ be a square matrix satisfying $A^{2} - 4 I = 0$ . Which of the following CANNOT be an eigenvalue of $A$ ?

A.
2
B.
-2
C.
0
D.
Both 2 and -2

Correct Answer:

Step-by-Step Solution

Key idea: This is a matrix polynomial question. Recognisable because it gives a polynomial equation that $A$ satisfies and asks about the possible eigenvalues.

Step 1: Recall the property of matrix polynomials. If a matrix $A$ satisfies $p (A) = 0$ , then every eigenvalue $λ$ of $A$ must be a root of the scalar polynomial $p (x) = 0$ .

Step 2: Identify the polynomial from the given equation.

The equation is $A^{2} - 4 I = 0$ . The corresponding scalar polynomial is:

$p (x) = x^{2} - 4$

Step 3: Find the roots of the polynomial.

Set $x^{2} - 4 = 0$ , which gives $x^{2} = 4$ .

The roots are $x = 2$ and $x = - 2$ .

Step 4: Determine the possible eigenvalues.

The only possible eigenvalues for $A$ are 2 and -2. Any other number cannot be an eigenvalue.

Step 5: Evaluate the options.

2 is a possible eigenvalue.
-2 is a possible eigenvalue.
0 is NOT a root of $x^{2} - 4 = 0$ , so it CANNOT be an eigenvalue.

Answer: Option C is correct.

Question 3 · Linear Algebra MCQ

If a square matrix $A$ satisfies the equation $A^{2} - 5 A + 6 I = 0$ , which of the following is a possible eigenvalue of $A$ ?

A.
$0$
B.
$5$
C.
$6$
D.
$2$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct application of the matrix polynomial eigenvalue property. Recognisable because it gives a polynomial equation that $A$ satisfies and asks for a possible eigenvalue.

Step 1: Recall the property that if a matrix $A$ satisfies $p (A) = 0$ , then every eigenvalue $λ$ of $A$ must be a root of the scalar polynomial $p (x) = 0$ .

Step 2: Identify the polynomial from the given equation.

The equation is $A^{2} - 5 A + 6 I = 0$ . The corresponding scalar polynomial is:

$p (x) = x^{2} - 5 x + 6$

Step 3: Find the roots of the polynomial by setting it to zero.

$x^{2} - 5 x + 6 = 0$

Factor the quadratic:

$(x - 2) (x - 3) = 0$

The roots are $x = 2$ and $x = 3$ .

Step 4: Determine the possible eigenvalues.

The only possible eigenvalues for $A$ are 2 and 3.

Step 5: Evaluate the options.

0 is not a root.
5 is not a root.
6 is not a root.
2 is a root, so it is a possible eigenvalue.

Answer: Option D is correct.

Question 4 · Linear Algebra MCQ

Let $A$ be a square matrix satisfying $A^{2} + 2 A = 0$ . If $λ$ is an eigenvalue of $A$ , which of the following could be a value of $λ$ ?

A.
1
B.
2
C.
-2
D.
3

Correct Answer:

Step-by-Step Solution

Key idea: This is a matrix polynomial question. When a matrix satisfies a polynomial equation $p (A) = 0$ , its eigenvalues must satisfy the same scalar polynomial equation $p (λ) = 0$ .

Step 1: The given matrix equation is $A^{2} + 2 A = 0$ .

Step 2: Replace $A$ with the scalar eigenvalue $λ$ to form the characteristic scalar equation: $λ^{2} + 2 λ = 0$ .

Step 3: Factor the equation: $λ (λ + 2) = 0$ .

Step 4: Solve for $λ$ : the possible eigenvalues are $λ = 0$ and $λ = - 2$ .

Step 5: Compare with the given options. The value $- 2$ is among the possible eigenvalues.

Answer: -2

Question 5 · Linear Algebra MCQ

Let $A$ be a square matrix satisfying the equation $A^{3} = A$ . If $λ$ is an eigenvalue of $A$ , which of the following scalar equations must $λ$ satisfy?

A.
$λ^{3} = 1$
B.
$λ^{2} = 1$
C.
$λ^{3} = λ$
D.
$λ^{3} = 0$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct pattern recognition question for matrix polynomials. Recognisable because it gives a simple polynomial equation $A^{3} = A$ and asks for the corresponding scalar equation for the eigenvalues.

Step 1: Recall the fundamental property of matrix polynomials. If a matrix $A$ satisfies a polynomial equation $p (A) = 0$ , then every eigenvalue $λ$ of $A$ must satisfy the exact same scalar polynomial equation $p (λ) = 0$ .

Step 2: Identify the polynomial equation from the given matrix equation. The equation is $A^{3} = A$ , which can be rewritten as $A^{3} - A = 0$ .

Step 3: Apply the property to the eigenvalues. Replace the matrix $A$ with the scalar eigenvalue $λ$ . The equation becomes $λ^{3} - λ = 0$ , or simply $λ^{3} = λ$ .

Step 4: Check the options. The equation $λ^{3} = λ$ matches Option C.

Answer: Option C is correct.

A question from this chapter

Question 1

For an m \times n matrix A, if the rank of A is strictly less than n, then the homogeneous system of linear equations A x = 0 has:

Question 2

Let A be a square matrix satisfying A^{2} - 4 I = 0 . Which of the following CANNOT be an eigenvalue of A ?

Question 3

If a square matrix A satisfies the equation A^{2} - 5 A + 6 I = 0, which of the following is a possible eigenvalue of A ?

Question 4

Let A be a square matrix satisfying A^{2} + 2 A = 0 . If λ is an eigenvalue of A, which of the following could be a value of λ ?

Question 5

Let A be a square matrix satisfying the equation A^{3} = A . If λ is an eigenvalue of A, which of the following scalar equations must λ satisfy?

Free preview ends here

Login to view the complete practice questions and solutions

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free

Rank, Invertibility and Linear Systems Practice Questions for GATE DA: 20+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Rank, Invertibility and Linear Systems

Chapter Roadmap

Rank, Nullity and the Solution Space

Rank, Nullity and the Solution Space

Rank, Invertibility and Linear Systems: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

More practice questions in this unit

A question from this chapter

Login to view the complete practice questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course