1
GATE CSE 2023
MCQ (Single Correct Answer)
+1
-0.33

The Lucas sequence $$L_n$$ is defined by the recurrence relation:

$${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$,

with $${L_1} = 1$$ and $${L_2} = 3$$.

Which one of the options given is TRUE?

A
$${L_n} = {\left( {{{1 + \sqrt 5 } \over 2}} \right)^n} + {\left( {{{1 - \sqrt 5 } \over 2}} \right)^n}$$
B
$${L_n} = {\left( {{{1 + \sqrt 5 } \over 2}} \right)^n} - {\left( {{{1 - \sqrt 5 } \over 3}} \right)^n}$$
C
$${L_n} = {\left( {{{1 + \sqrt 5 } \over 2}} \right)^n} + {\left( {{{1 - \sqrt 5 } \over 3}} \right)^n}$$
D
$${L_n} = {\left( {{{1 + \sqrt 5 } \over 2}} \right)^n} - {\left( {{{1 - \sqrt 5 } \over 2}} \right)^n}$$
2
GATE CSE 2023
MCQ (Single Correct Answer)
+1
-0.33

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \right]$$.

Let $$\mathrm{det}(A)$$ and $$\mathrm{det}(B)$$ denote the determinates of the matrices A and B, respectively.

Which one of the options given below is TRUE?

A
$$\mathrm{det}(A)=\mathrm{det}(B)$$
B
$$\mathrm{det}(B)=-\mathrm{det}(A)$$
C
$$\mathrm{det}(A)=0$$
D
$$\mathrm{det}(AB)=\mathrm{det}(A)+\mathrm{det}(B)$$
3
GATE CSE 2023
Numerical
+1
-0

Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.

GATE CSE 2023 Discrete Mathematics - Linear Algebra Question 12 English

Let $$\lambda_1,\lambda_2,\lambda_3,\lambda_4$$, and $$\lambda_5$$ be the five eigenvalues of A. Note that these eigenvalues need not be distinct.

The value of $$\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5=$$ ______________

Your input ____
4
GATE CSE 2022
MCQ (Single Correct Answer)
+1
-0.33

Consider the following two statements with respect to the matrices Am $$\times$$ n , Bn $$\times$$ m , Cn$$\times$$ n and Dn $$\times$$ n .

Statement 1 : tr(AB) = tr(BA)

Statement 2 : tr(CD) = tr(DC)

where tr( ) represents the trace of a matrix. Which one of the following holds?

A
Statement 1 is correct and Statement 2 is wrong.
B
Statement 1 is wrong and Statement 2 is correct.
C
Both Statement 1 and Statement 2 are correct.
D
Both Statement 1 and Statement 2 are wrong.
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