1
GATE EE 2016 Set 2
+2
-0.6
Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{ x \cr y \cr } } \right)$$ such that $${a^2} + {b^2} = 1$$ where $$\left( {\matrix{ a \cr b \cr } } \right) = P\left( {\matrix{ x \cr y \cr } } \right).$$ Then $$S$$ is
A
a circle of radius $$\sqrt {10}$$
B
a circle of radius $${1 \over {\sqrt {10} }}$$
C
an ellipse with major axis along $$\left( {\matrix{ 1 \cr 1 \cr } } \right)$$
D
an ellipse with minor axis along $$\left( {\matrix{ 1 \cr 1 \cr } } \right)$$
2
GATE EE 2016 Set 1
+2
-0.6
Let $$A$$ be a $$4 \times 3$$ real matrix which rank$$2.$$ Which one of the following statement is TRUE?
A
Rank of AT is less than $$2$$
B
Rank of ATA is equal to $$2$$
C
Rank of ATA is greater than $$2$$
D
Rank of ATA can be any number between $$1$$ and $$3$$
3
GATE EE 2016 Set 1
+2
-0.6
Let the eigenvalues of a $$2 \times 2$$ matrix $$A$$ be $$1,-2$$ with eigenvectors $${x_1}$$ and $${x_2}$$ respectively. Then the eigenvalues and eigenvectors of the matrix $${A^2} - 3A + 4{\rm I}$$ would respectively, be
A
$$2,14;{\,x_1},{x_2}$$
B
$$2,14;{x_1} + {x_2}:{x_1} - {x_2}$$
C
$$2,0;{\,x_1},{x_2}$$
D
$$2,0;\,{x_1} + {x_2},\,{x_1} - {x_2}$$
4
GATE EE 2015 Set 1
+2
-0.6
The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{ { - 3} & 0 & { - 2} \cr 1 & { - 1} & 0 \cr 0 & a & { - 2} \cr } } \right]$$ has three linearly independent real eigenvectors is
A
$${2 \over {3\sqrt 3 }}$$
B
$${1 \over {3\sqrt 3 }}$$
C
$${{1 + 2\sqrt 3 } \over {3\sqrt 3 }}$$
D
$${{1 + \sqrt 3 } \over {3\sqrt 3 }}$$
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