1
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 $$\left| {\left| {P\overline x } \right|} \right|$$
2
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
$$n$$
3
GATE EE 2007
+2
-0.6
If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]\,$$ then $${A^9}$$ equals
A
$$511\,\,A + 510\,\,I$$
B
$$309\,\,A + 104\,\,I$$
C
$$154\,\,A + 155\,\,I$$
D
$${e^{9A}}$$
4
GATE EE 2007
+2
-0.6
If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]$$ then $$A$$ satisfies the relation
A
$$A - 31 + 2\,{A^{ - 1}} = 0$$
B
$${A^2} + 2A + 2I = 0$$
C
$$\left( {A + I} \right)\left( {A + 2I} \right) = 0$$
D
$${e^A} = 0$$
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