Ee gate 2008

Question

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

A.  $||P{\vec{x}}||$ $\leq$ $||{\vec{x}}||$ where at least one vector satisfies $||P{\vec{x}}||$ <$||{\vec{x}}||$

B.  $||P{\vec{x}}||$ = $||{\vec{x}}||$ for all vectors ${\vec{x}}$

C.  $||P{\vec{x}}||$ $\geq$ $||{\vec{x}}||$ where at least one vector satisfies $||P{\vec{x}}||$ >$||{\vec{x}}||$

D.  No relationship can be established between$||{\vec{x}}||$ and $||P{\vec{x}}||$

Answer 1

By my approach Answer is c option -

Solution -

Comments

and what is real matrix, real vector meaning here?

is it means it is only for real numbers?

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Yes it is for real numbers.

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Can u give any relevant reference link for this?

matrix will be

$\begin{bmatrix} a& c \end{bmatrix}$$\begin{bmatrix} b\\d \end{bmatrix}$

Instead of what u write