1
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
2
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.
3
GATE EE 2008
+2
-0.6
Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x$$ is a real vector $${\left[ {\matrix{ {{x_1}} & {{x_2}} \cr } } \right]^T}$$ with length $$\left| {\left| {\overline x } \right|} \right| = {\left( {{x_1}^2 + {x_2}^2} \right)^{1/2}}.$$ Then which one of the following statement is correct?
A
$$\left| {\left| {P\overline x } \right|} \right| \le \left| {\left| {\overline x } \right|} \right|$$ where at least one vector satisfies $$\left| {\left| {P\overline x } \right|} \right| < \left| {\left| {\overline x } \right|} \right|$$
B
$$\left| {\left| {P\overline x } \right|} \right| = \left| {\left| {\overline x } \right|} \right|$$ for all vectors $${\overline x }$$
C
$$\left| {\left| {P\overline x } \right|} \right| \ge \left| {\left| {\overline x } \right|} \right|$$ where at least one vector satisfies $$\left| {\left| {P\overline x } \right|} \right| > \left| {\left| {\overline x } \right|} \right|$$
D
No relationship can be established between $$\left| {\left| {\overline x } \right|} \right|$$ and $$\left| {\left| {P\overline x } \right|} \right|$$
4
GATE EE 2007
+2
-0.6
$${q_1},\,{q_2},{q_3},.......{q_m}$$ are $$n$$-dimensional vectors with $$m < n.$$ This set of vectors is linearly dependent. $$Q$$ is the matrix with $${q_1},\,{q_2},{q_3},.......{q_m}$$ as the columns. The rank of $$Q$$ is
A
less than $$m$$
B
$$m$$
C
between $$m$$ and $$n$$
D
$$n$$
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NEET