GATE CSE 2014 Set 3 | Linear Algebra Question 52 | Discrete Mathematics

GATE CSE 2014 Set 3

Numerical

-0

If $${V_1}$$ and $${V_2}$$ are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of $${V_1}\, \cap \,\,{V_2}$$ is _________________.

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GATE CSE 2014 Set 3

MCQ (Single Correct Answer)

-0.3

Which one of the following statements is TRUE about every $$n\,\, \times \,n$$ matrix with only real eigen values?

If the trace of the matrix is positive and the determinant of the negative, at least one of its eigen values is negative.

If the trace of the matrix is positive, all its eigen values are positive.

If the determinanant of the matrix is positive, all its eigen values are positive.

If the product of the trace and determination of the matrix is positive, all its eigen values are positive.

GATE CSE 2013

MCQ (Single Correct Answer)

-0.3

Which of the following does not equal
$$\left| {\matrix{ 1 & x & {{x^2}} \cr 1 & y & {{y^2}} \cr 1 & z & {{z^2}} \cr } } \right|?$$

$$\left| {\matrix{ 1 & {x\left( {x + 1} \right)} & {x + 1} \cr 1 & {y\left( {y + 1} \right)} & {y + 1} \cr 1 & {z\left( {z + 1} \right)} & {z + 1} \cr } } \right|$$

$$\left| {\matrix{ 1 & {x + 1} & {{x^2} + 1} \cr 1 & {y + 1} & {{y^2} + 1} \cr 1 & {z + 1} & {{z^2} + 1} \cr } } \right|$$

$$\left| {\matrix{ 0 & {x - y} & {{x^2} - {y^2}} \cr 0 & {y - z} & {{x^2} - {z^2}} \cr 1 & z & {{z^2}} \cr } } \right|$$

$$\left| {\matrix{ 2 & {x + y} & {{x^2} + {y^2}} \cr 2 & {y + z} & {{x^2} + {z^2}} \cr 1 & z & {{z^2}} \cr } } \right|$$

GATE CSE 2012

MCQ (Single Correct Answer)

-0.3

Let $$A$$ be the $$2 \times 2$$ matrix with elements $${a_{11}} = {a_{12}} = {a_{21}} = + 1$$ and $${a_{22}} = - 1$$. Then the eigen values of the matrix $${A^{19}}$$ are

$$1024$$ and $$-1024$$

$$1024\sqrt 2 \,i$$ and $$ - 1024\sqrt 2 $$

$$4\sqrt 2 $$ and $$-4\sqrt 2 $$

$$512\sqrt 2 $$ and $$-512\sqrt 2 $$

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