1
GATE EE 2008
+1
-0.3
The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as
$$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda + {\lambda ^2} + 1 = 0.$$
If $${\rm I}$$ denotes identity matrix then the inverse of $$P$$ will be
A
$${P^2} + P + 2{\rm I}$$
B
$${P^2} + P + {\rm I}$$
C
$$- \left( {{P^2} + P + {\rm I}} \right)$$
D
$$- \left( {{P^2} + P + 2{\rm I}} \right)$$
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 2007
+1
-0.3
$$X = {\left[ {\matrix{ {{x_1}} & {{x_2}} & {.......\,{x_n}} \cr } } \right]^T}$$ is an $$n$$-tuple non-
zero vector. The $$n\,\, \times \,\,n$$ matrix $$V = X{X^T}$$
A
has rank zero
B
has rank $$1$$
C
is orthogonal
D
has rank $$n$$
4
GATE EE 2005
+1
-0.3
In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solution for the unknown vector $$X.$$
A
Augmented matrix $$\left[ {P|Q} \right]$$ must have the same rank as matrix $$P.$$
B
vector $$Q$$ must have only non-zero elements.
C
matrix $$P$$ must be singular
D
matrix $$P$$ must be square
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