The identity element in the group M = . ( * 20 c x&x &x ) |x R; ,… - Vidyadip

MCQ

The identity element in the group $M = \left\{ {\left. {\left( {\begin{array}{*{20}{c}}x&x\\x&x\end{array}} \right)} \right|x \in R;\,x \ne 0\,} \right\}$ with respect to matrix multiplication is

A
$\left( {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right)$
✓
$\frac{1}{2}\left( {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right)$
C
$\left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right)$
D
$\left( {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right)$

✓

Answer

Correct option: B.

$\frac{1}{2}\left( {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right)$

b
(b) Let $\left[ {\begin{array}{*{20}{c}}a&a\\a&a\end{array}} \right]$be the identity element then

$\left[ {\begin{array}{*{20}{c}}x&x\\x&x\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a&a\\a&a\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}x&x\\x&x\end{array}} \right]$

i.e., $2ax = x \Rightarrow a = \frac{1}{2}$,

Identity element = $\frac{1}{2}\left[ {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right]$.

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