GATE ECE 2023 | Linear Algebra Question 3 | Engineering Mathematics

GATE ECE 2023

MCQ (Single Correct Answer)

-0.33

Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are

$$\{ {\lambda _k}\,\mathrm{det}(A)|1 \le k \le n\} $$ and $$\{ P{v_k}|1 \le k \le n\} $$

$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$

$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ P{v_k}|1 \le k \le n\} $$

$$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {P^{ - 1}}{v_k}|1 \le k \le n\} $$

GATE ECE 2022

MCQ (Single Correct Answer)

-0.33

Consider a system of linear equations Ax = b, where

$$A = \left[ {\matrix{ 1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr { - 1} \hfill & {\sqrt 2 } \hfill & { - 3} \hfill \cr } } \right]$$, $$b = \left[ {\matrix{ 1 \cr 3 \cr } } \right]$$

This system is equations admits __________.

a unique solution for x

infinitely many solutions for x

no solutions for x

exactly two solutions for x

GATE ECE 2018

Numerical

-0.33

Consider matrix $$A = \left[ {\matrix{ k & {2k} \cr {{k^2} - k} & {{k^2}} \cr } } \right]$$ and

vector $$X = \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$.

The number of distinct real values of k for which the equation AX = 0 has infinitely many solutions is _______.

Your input ____

GATE ECE 2018

MCQ (Single Correct Answer)

-0.33

Let M be a real 4 $$ \times $$ 4 matrix. Consider the following statements :

S1: M has 4 linearly independent eigenvectors.

S2: M has 4 distinct eigenvalues.

S3: M is non-singular (invertible).

Which one among the following is TRUE?

S1 implies S2

S2 implies S1

S1 implies S3

S3 implies S2

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