GATE ECE 2023 | Linear Algebra Question 4 | Engineering Mathematics

GATE ECE 2023

MCQ (Single Correct Answer)

-0.67

Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is

$${l^2}$$

$${{{l^2}} \over 4}$$

$$l$$

$${{{l^2}} \over 2}$$

GATE ECE 2023

MCQ (Single Correct Answer)

-0.67

The state equation of a second order system is

$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.

Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and $$v_1$$ and $$v_2$$ are the corresponding eigenvectors. For constants $$\alpha_1$$ and $$\alpha_2$$, the solution, $$x(t)$$, of the state equation is

$$\sum\limits_{i = 1}^2 {{\alpha _i}{e^{{\lambda _i}t}}{v_i}} $$

$$\sum\limits_{i = 1}^2 {{\alpha _i}{e^{2{\lambda _i}t}}{v_i}} $$

$$\sum\limits_{i = 1}^2 {{\alpha _i}{e^{3{\lambda _i}t}}{v_i}} $$

$$\sum\limits_{i = 1}^2 {{\alpha _i}{e^{4{\lambda _i}t}}{v_i}} $$

GATE ECE 2022

MCQ (Single Correct Answer)

-0.67

Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v₁ and v₂ satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :

A = $$\alpha$$v₁$$v_1^T$$ + $$\beta$$v₂$$v_2^T$$

The eigen values of A are __________.

0, $$\alpha$$, $$\beta$$

0, $$\alpha$$ + $$\beta$$, $$\alpha$$ $$-$$ $$\beta$$

0, $${{\alpha + \beta } \over 2},\sqrt {\alpha \beta } $$

0, 0, $$\sqrt {{\alpha ^2} + {\beta ^2}} $$

GATE ECE 2017 Set 2

Numerical

-0

The rank of the matrix $$\left[ {\matrix{ 1 & { - 1} & 0 & 0 & 0 \cr 0 & 0 & 1 & { - 1} & 0 \cr 0 & 1 & { - 1} & 0 & 0 \cr { - 1} & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 1 & { - 1} \cr } } \right]$$ is __________.

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