1
GATE IN 2016
Numerical
+2
-0
Consider the matrix $$A = \left( {\matrix{ 2 & 1 & 1 \cr 2 & 3 & 4 \cr { - 1} & { - 1} & { - 2} \cr } } \right)$$ whose eigen values are $$1, -1$$ and $$3$$. Then trace of $$\left( {{A^3} - 3{A^2}} \right)$$ is ________.
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2
GATE IN 2013
MCQ (Single Correct Answer)
+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
$$\left[ {\matrix{ 1 \cr j \cr } } \right],\left[ {\matrix{ j \cr 1 \cr } } \right]$$
3
GATE IN 2007
MCQ (Single Correct Answer)
+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
4
GATE IN 2006
MCQ (Single Correct Answer)
+2
-0.6
A system of linear simultaneous equations is given as $$AX=b$$
where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]$$

Which of the following statement is true?

A
$$x$$ is a null vector
B
$$x$$ is unique
C
$$x$$ does not exist
D
$$x$$ has infinitely many values
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