1
GATE ECE 2022
MCQ (Single Correct Answer)
+1
-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
a unique solution for x
B
infinitely many solutions for x
C
no solutions for x
D
exactly two solutions for x
2
GATE ECE 2022
MCQ (Single Correct Answer)
+1
-0.33

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 __________.

A
0, $$\alpha$$, $$\beta$$
B
0, $$\alpha$$ + $$\beta$$, $$\alpha$$ $$-$$ $$\beta$$
C
0, $${{\alpha + \beta } \over 2},\sqrt {\alpha \beta } $$
D
0, 0, $$\sqrt {{\alpha ^2} + {\beta ^2}} $$
3
GATE ECE 2018
MCQ (Single Correct Answer)
+1
-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?
A
S1 implies S2
B
S2 implies S1
C
S1 implies S3
D
S3 implies S2
4
GATE ECE 2018
Numerical
+1
-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 _______.
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