1
GATE IN 2013
+2
-0.6
One pair of eigenvectors corresponding to the two eigen values of the matrix $$\left[ {\matrix{ 0 & { - 1} \cr 1 & {0 - } \cr } } \right]$$
A
$$\left[ {\matrix{ 1 \cr { - j} \cr } } \right],\left[ {\matrix{ j \cr { - 1} \cr } } \right]$$
B
$$\left[ {\matrix{ 0 \cr 1 \cr } } \right],\left[ {\matrix{ { - 1} \cr 0 \cr } } \right]$$
C
$$\left[ {\matrix{ 1 \cr j \cr } } \right],\left[ {\matrix{ 0 \cr 1 \cr } } \right]$$
D
$$\left[ {\matrix{ 1 \cr j \cr } } \right],\left[ {\matrix{ j \cr 1 \cr } } \right]$$
2
GATE IN 2007
+2
-0.6
Let $$A$$ be $$n \times n$$ real matrix such that $${A^2} = {\rm I}$$ and $$Y$$ be an $$n$$-diamensional vector. Then the linear system of equations $$Ax=y$$ has
A
no solution
B
unique solution
C
more than one but infinitely many dependent solutions.
D
Infinitely many dependent solutions
3
GATE IN 2006
+2
-0.6
For a given $$2x2$$ matrix $$A,$$ it is observved that $$A\left[ {\matrix{ 1 \cr { - 1} \cr } } \right] = - 1\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$ and
$$A\left[ {\matrix{ 1 \cr { - 2} \cr } } \right] = - 2\left[ {\matrix{ 1 \cr { - 2} \cr } } \right]$$ then the matrix $$A$$ is
A
$$A = \left[ {\matrix{ 1 & 1 \cr { - 1} & { - 2} \cr } } \right]\,\,\left[ {\matrix{ 1 & 0 \cr 1 & 2 \cr } } \right]\,\,\left[ {\matrix{ 2 & 1 \cr { - 1} & { - 1} \cr } } \right]$$
B
$$A = \left[ {\matrix{ 1 & 1 \cr { - 1} & { - 2} \cr } } \right]\,\,\left[ {\matrix{ 1 & 0 \cr 1 & 2 \cr } } \right]\,\,\left[ {\matrix{ 2 & 1 \cr { - 1} & { - 1} \cr } } \right]$$
C
$$A = \left[ {\matrix{ 1 & 1 \cr { - 1} & { - 2} \cr } } \right]\,\,\left[ {\matrix{ { - 1} & 0 \cr 0 & { - 2} \cr } } \right]\,\,\left[ {\matrix{ 2 & 1 \cr { - 1} & { - 1} \cr } } \right]$$
D
$$A = \left[ {\matrix{ 0 & { - 2} \cr 1 & { - 3} \cr } } \right]$$
4
GATE IN 2006
+2
-0.6
A system of linear simultaneous equations is given as $$AX=b$$
where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]$$

Then the rank of matrix $$A$$ is

A
$$1$$
B
$$2$$
C
$$3$$
D
$$4$$
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