GATE IN 2006 | Linear Algebra Question 17 | Engineering Mathematics

1

GATE IN 2006

MCQ (Single Correct Answer)

+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

$$A = \left[ {\matrix{ 0 & { - 2} \cr 1 & { - 3} \cr } } \right]$$

2

GATE IN 2006

MCQ (Single Correct Answer)

+2

-0.6

A system of linear simultaneous equations is given as $$AX=b$$
where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 & 1 \cr 1 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right]\,\,\& \,\,b = \left[ {\matrix{ 0 \cr 0 \cr 0 \cr 1 \cr } } \right]$$

Then the rank of matrix $$A$$ is

A

$$1$$

B

$$2$$

C

$$3$$

D

$$4$$

3

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