1
GATE IN 2012
+1
-0.3
Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the value of $${A^3}$$ is
A
$$15A+12$$ $${\rm I}$$
B
$$19A+30$$ $${\rm I}$$
C
$$17A+15$$ $${\rm I}$$
D
$$17A+21$$ $${\rm I}$$
2
GATE IN 2011
+1
-0.3
The matrix $$M = \left[ {\matrix{ { - 2} & 2 & { - 3} \cr 2 & 1 & 6 \cr { - 1} & { - 2} & 0 \cr } } \right]$$ has eigen values $$-3, -3, 5.$$ An eigen vector corresponding to the eigen value $$5$$ is $${\left[ {\matrix{ 1 & 2 & { - 1} \cr } } \right]^T}.$$ One of the eigen vector of the matrix $${M^3}$$ is
A
$${\left[ {\matrix{ 1 & 8 & { - 1} \cr } } \right]^T}$$
B
$${\left[ {\matrix{ 1 & 2 & { - 1} \cr } } \right]^T}$$
C
$${\left[ {\matrix{ 1 & {\root 3 \of 2 } & { - 1} \cr } } \right]^T}$$
D
$${\left[ {\matrix{ 1 & 1 & { - 1} \cr } } \right]^T}$$
3
GATE IN 2010
+1
-0.3
$$X$$ and $$Y$$ are non-zero square matrices of size $$n \times n$$. If $$XY = {O_{n \times n}}$$ then
A
$$\left| X \right| = 0\,$$ and $$\,\left| Y \right| \ne 0$$
B
$$\left| X \right| \ne 0\,$$ and $$\,\left| Y \right| = 0$$
C
$$\left| X \right| = 0$$ and $$\,\left| Y \right| = 0$$
D
$$\left| X \right| \ne 0$$ and $$\,\left| Y \right| \ne 0$$
4
GATE IN 2010
+1
-0.3
A real $$n \times n$$ matrix $$A$$ $$= \left[ {{a_{ij}}} \right]$$ is defined as
follows $$\left\{ {\matrix{ {{a_{ij}} = i,} & {\forall i = j} \cr { = 0,} & {otherwise} \cr } .} \right.$$

The sum of all $$n$$ eigen values of $$A$$ is

A
$${{n\left( {n + 1} \right)} \over 2}$$
B
$${{n\left( {n - 1} \right)} \over 2}$$
C
$${{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 2}$$
D
$${{n^2}}$$
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