GATE EE 2011 | Linear Algebra Question 22 | Engineering Mathematics

GATE EE 2011

MCQ (Single Correct Answer)

-0.6

The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a product of lower triangular matrix $$\left[ L \right]$$ and an upper triangular $$\left[ U \right].$$ The properly decomposed $$\left[ L \right]$$ and $$\left[ U \right]$$ matrices respectively are

$$\left[ {\matrix{ 1 & 0 \cr 4 & { - 1} \cr } } \right]$$ and $$\left[ {\matrix{ 1 & 1 \cr 0 & { - 2} \cr } } \right]$$

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

$$\left[ {\matrix{ 1 & 0 \cr 4 & 1 \cr } } \right]\,$$ and $$\left[ {\matrix{ 2 & 1 \cr 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ 2 & 0 \cr 4 & { - 3} \cr } } \right]$$ and $$\left[ {\matrix{ 1 & {0.5} \cr 0 & 1 \cr } } \right]$$

GATE EE 2010

MCQ (Single Correct Answer)

-0.6

For the set of equations $$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$ $$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$
The following statement is true

only the trivial solution $${x_1} = {x_2} = {x_3} = {x_4} = 0$$ exist

there are no solutions

a unique non-trivial solution exist

multiple non-trivial solution exist

GATE EE 2008

MCQ (Single Correct Answer)

-0.6

If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?

$$Q$$ will have four linearly independent rows and four linearly independent columns

$$Q$$ will have four linearly independent rows and five linearly independent columns

$$Q{Q^T}$$ will be invertible.

$${Q^T}Q$$ will be invertible.

GATE EE 2008

MCQ (Single Correct Answer)

-0.6

Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x $$ is a real vector $${\left[ {\matrix{ {{x_1}} & {{x_2}} \cr } } \right]^T}$$ with length $$\left| {\left| {\overline x } \right|} \right| = {\left( {{x_1}^2 + {x_2}^2} \right)^{1/2}}.$$ Then which one of the following statement is correct?

$$\left| {\left| {P\overline x } \right|} \right| \le \left| {\left| {\overline x } \right|} \right|$$ where at least one vector satisfies $$\left| {\left| {P\overline x } \right|} \right| < \left| {\left| {\overline x } \right|} \right|$$

$$\left| {\left| {P\overline x } \right|} \right| = \left| {\left| {\overline x } \right|} \right|$$ for all vectors $${\overline x }$$

$$\left| {\left| {P\overline x } \right|} \right| \ge \left| {\left| {\overline x } \right|} \right|$$ where at least one vector satisfies $$\left| {\left| {P\overline x } \right|} \right| > \left| {\left| {\overline x } \right|} \right|$$

No relationship can be established between $$\left| {\left| {\overline x } \right|} \right|$$ and $$\left| {\left| {P\overline x } \right|} \right|$$

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