1
GATE EE 2014 Set 3
+2
-0.6
$$A = \left[ {\matrix{ p & q \cr r & s \cr } } \right];B = \left[ {\matrix{ {{p^2} + {q^2}} & {pr + qs} \cr {pr + qs} & {{r^2} + {s^2}} \cr } } \right]$$
If the rank of matrix $$A$$ is $$N$$, then the rank of matrix $$B$$ is
A
$$N/2$$
B
$$N-1$$
C
$$N$$
D
$$2$$ $$N$$
2
GATE EE 2013
+2
-0.6
The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr } } \right]$$ has
A
no solution
B
only one solution
C
non-zero unique solution
D
multiple solutions
3
GATE EE 2013
+2
-0.6
A matrix has eigen values $$-1$$ and $$-2.$$ The corresponding eigenvectors are $$\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$ and $$\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]$$ respectively. The matrix is
A
$$\left[ {\matrix{ 1 & 1 \cr { - 1} & { - 2} \cr } } \right]$$
B
$$\left[ {\matrix{ 1 & 2 \cr { - 2} & { - 4} \cr } } \right]$$
C
$$\left[ {\matrix{ { - 1} & 0 \cr 0 & { - 2} \cr } } \right]$$
D
$$\left[ {\matrix{ 0 & 1 \cr { - 2} & { - 3} \cr } } \right]$$
4
GATE EE 2011
+2
-0.6
The two vectors $$\left[ {\matrix{ 1 & 1 & 1 \cr } } \right]$$ and $$\left[ {\matrix{ 1 & a & {{a^2}} \cr } } \right]$$ where $$a = - {1 \over 2} + j{{\sqrt 3 } \over 2}$$ and $$j = \sqrt { - 1}$$ are
A
orthonormal
B
orthogonal
C
parallel
D
collinear
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