1
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
$${e^{9A}}$$
2
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
$${e^A} = 0$$
3
GATE EE 2007
+2
-0.6
Let $$x$$ and $$y$$ be two vectors in a $$3-$$ dimensional space and $$< x,y >$$ denote their dot product. Then the determinant det $$\left[ {\matrix{ { < x,x > } & { < x,y > } \cr { < y,x > } & { < y,y > } \cr } } \right] =$$ ______.
A
is zero when $$x$$ and $$y$$ are linearly independent
B
is positive when $$x$$ and $$y$$ are linearly independent
C
is non-zero for all non-zero $$x$$ and $$y$$
D
is zero only when either $$x$$ (or) $$y$$ is zero
4
GATE EE 2005
+2
-0.6
For the matrix $$P = \left[ {\matrix{ 3 & { - 2} & 2 \cr 0 & { - 2} & 1 \cr 0 & 0 & 1 \cr } } \right],$$ one of the eigen values is $$-2.$$ Which of the following is an eigen vector?
A
$$\left( {\matrix{ 3 \cr { - 2} \cr 1 \cr } } \right)$$
B
$$\left[ {\matrix{ { - 3} \cr 2 \cr { - 1} \cr } } \right]$$
C
$$\left[ {\matrix{ 1 \cr { - 2} \cr 3 \cr } } \right]$$
D
$$\left[ {\matrix{ 2 \cr 5 \cr 0 \cr } } \right]$$
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