1
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.
2
GATE EE 2008
+1
-0.3
$$A$$ is $$m$$ $$x$$ $$n$$ full rank matrix with $$m > n$$ and $${\rm I}$$ is an identity matrix. Let matrix $${A^ + } = {\left( {{A^T}A} \right)^{ - 1}}{A^T}.$$ Then which one of the following statement is false?
A
$$A{A^ + }A = A$$
B
$${\left( {A{A^ + }} \right)^2} = A{A^ + }$$
C
$${A^ + }A = {\rm I}$$
D
$$A{A^ + }A = {A^ + }$$
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 2008
+1
-0.3
$$X$$ is uniformly distributed random variable that take values between $$0$$ and $$1.$$ The value of $$E\left( {{X^3}} \right)$$ will be
A
$$0$$
B
$$1/8$$
C
$$1/4$$
D
$$1/2$$
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