GATE CSE 2024 Set 1 | Linear Algebra Question 3 | Discrete Mathematics

GATE CSE 2024 Set 1

MCQ (More than One Correct Answer)

-0.66

Let A be any n x m matrix, where m > n. Which of the following statements is/are TRUE about the system of linear equations Ax = 0?

There exist at least m - n linearly independent solutions to this system

There exist m - n linearly independent vectors such that every solution is a linear combination of these vectors

There exists a non-zero solution in which at least m - n variables are 0

There exists a solution in which at least n variables are non-zero

GATE CSE 2021 Set 1

Numerical

-0.67

Consider the following matrix.

$$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{array}} \right)$$

The largest eigenvalue of the above matrix is ______

Your input ____

GATE CSE 2020

MCQ (Single Correct Answer)

-0.67

Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements,

I. rank(AB) = rank(A) rank(B)
II. det(AB) = det(A) det(B)
III. rank(A + B) $$ \le $$ rank(A) + rank(B)
IV. det(A + B) $$ \le $$ det(A) + det(B)

Which of the above statements are TRUE?

I and II only

II and III only

I and IV only

III and IV only

GATE CSE 2018

MCQ (Single Correct Answer)

-0.6

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + 3$$ for all $$n = 0,1,2,....?$$

$${3 \over {{{\left( {1 - x} \right)}^2}}}$$

$${{3x} \over {{{\left( {1 - x} \right)}^2}}}$$

$${\left( {1 - x} \right)}$$

$${{3 - x} \over {{{\left( {1 - x} \right)}^2}}}$$

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