Rank, Invertibility and Linear Systems Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Rank, Invertibility and Linear Systems short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Rank, Nullity and Polynomials

1. Fundamental Theorem

Rank (A) + Nullity (A) = n

2. Matrix Polynomials $p (A) = 0$

Eigenvalues $λ$ must satisfy $p (λ) = 0$ .
If roots of $p (x)$ are distinct, $A$ is diagonalizable.

3. Special Matrices

Idempotent ( $A^{2} = A$ ): $λ \in {0, 1}$ , $Tr (A) = Rank (A)$ .
Nilpotent ( $A^{k} = 0$ ): $λ = 0$ , $det (A) = 0$ .
Involutory ( $A^{2} = I$ ): $λ \in {1, - 1}$ .

Strategy: Always check eigenvalues first. If the question asks about invertibility, check if 0 is a possible eigenvalue.

Summary: Rank-One Update Cheat Sheet

Rank-One Update Cheat Sheet

For $M = A + u v^{T}$ (where $A$ is invertible)

Determinant: $det (M) = det (A) (1 + v^{T} A^{- 1} u)$
Inverse: $M^{- 1} = A^{- 1} - \frac{A ^{- 1} u v ^{T} A ^{- 1}}{1 + v ^{T} A ^{- 1} u}$
Invertible if: $1 + v^{T} A^{- 1} u \neq = 0$

Special case: $A = I$

$det (I + u v^{T}) = 1 + v^{T} u$
$(I + u v^{T})^{- 1} = I - \frac{u v ^{T}}{1 + v ^{T} u}$

Eigenvalues of $u v^{T}$

v^{T} u

Mult. 1

0

Mult.

n - 1

Rank, Invertibility and Linear Systems: Solved Questions with Step-by-Step Explanations (2 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.

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Rank, Invertibility and Linear Systems Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Rank, Invertibility and Linear Systems short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

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 ?

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Rank, Invertibility and Linear Systems Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Summary: Rank, Nullity and Polynomials

Summary: Rank, Nullity and Polynomials

Summary: Rank-One Update Cheat Sheet

Rank-One Update Cheat Sheet

For M=A+uvT (where A is invertible)

Special case: A=I

Eigenvalues of uvT

Rank, Invertibility and Linear Systems: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

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Questions calibrated to the real exam

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Unlock the whole course

For $M = A + u v^{T}$ (where $A$ is invertible)

Special case: $A = I$

Eigenvalues of $u v^{T}$