GATE EC || ENGINEERING MATHE || LINEAR ALGEBRA || PYQs ( 2000-2025)

Consider the matrix A below:

A = [ 2 3 4 5

0 6 7 8

0 0 α β

0 0 0 γ ]

For which of the following combinations of α, β and γ is the rank of A at least three?

(i) α = 0 and β = γ ≠ 0

(ii) α = β = γ = 0

(iii) β = γ = 0 and α ≠ 0

(iv) α = β = γ ≠ 0

Let ℝ and ℝ³ denote the set of real numbers and the three-dimensional vector space over it, respectively. The value of α for which the set of vectors

{(2, −3α), (3, −1, 3), (1, −5, 7)} does not form a basis of ℝ³ is ________.

(GATE 2024 || EC || NAT ||1 MARK)

Let the sets of eigenvalues and eigenvectors of a matrix B be
{λk∣1≤k≤n} and {vk∣1≤k≤n}, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where B=P⁻¹AP,respectively, are __________.

(GATE 2023 || EC || MCQ ||1 MARK)

Consider a system of linear equations Ax = b, where

This system of equations admits __________.

(GATE 2022 || EC || MCQ ||1 MARK)

Consider the vectors

For real-valued scalar variable x, the value of

minₓ ||ax − b||₂ is __________ (rounded off to two decimal places). || · ||₂ denotes the Euclidean norm, i.e., for

(GATE 2025 || EC || NAT ||1 MARK)

Consider the matrix [1 k] [2​ 1​], where k is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

(GATE 2024 || EC || MCQ ||1 MARK)

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.

What is the probability that the two balls drawn have different colours?

( GATE 2025 || EC || MCQ ||1 MARK)

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable X  denote the sum of the outcomes obtained. The expectation of  is ___________ (rounded off to two decimal places)

(GATE 2025 || EC || NAT ||1 MARK)

Suppose  X and  Y are independent and identically distributed random variables that are distributed uniformly in the interval [0, 1] . The probability that X is greater than equal to Y is __

(GATE 2024 || EC || NAT ||1 MARK)

The bar graph shown the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is ____________ (rounded off to one decimal place).