1
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
2
GATE EE 2011
+2
-0.6
The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a product of lower triangular matrix $$\left[ L \right]$$ and an upper triangular $$\left[ U \right].$$ The properly decomposed $$\left[ L \right]$$ and $$\left[ U \right]$$ matrices respectively are
A
$$\left[ {\matrix{ 1 & 0 \cr 4 & { - 1} \cr } } \right]$$ and $$\left[ {\matrix{ 1 & 1 \cr 0 & { - 2} \cr } } \right]$$
B
$$\left[ {\matrix{ 1 & 0 \cr 2 & 1 \cr } } \right]$$ and $$\left[ {\matrix{ 2 & 1 \cr 0 & { - 3} \cr } } \right]$$
C
$$\left[ {\matrix{ 1 & 0 \cr 4 & 1 \cr } } \right]\,$$ and $$\left[ {\matrix{ 2 & 1 \cr 0 & { - 1} \cr } } \right]$$
D
$$\left[ {\matrix{ 2 & 0 \cr 4 & { - 3} \cr } } \right]$$ and $$\left[ {\matrix{ 1 & {0.5} \cr 0 & 1 \cr } } \right]$$
3
GATE EE 2010
+2
-0.6
For the set of equations $${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$
The following statement is true
A
only the trivial solution $${x_1} = {x_2} = {x_3} = {x_4} = 0$$ exist
B
there are no solutions
C
a unique non-trivial solution exist
D
multiple non-trivial solution exist
4
GATE EE 2008
+2
-0.6
If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?
A
$$Q$$ will have four linearly independent rows and four linearly independent columns
B
$$Q$$ will have four linearly independent rows and five linearly independent columns
C
$$Q{Q^T}$$ will be invertible.
D
$${Q^T}Q$$ will be invertible.
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