Question

prove that a^2 is symmetric if either a is symmetric or a is skew symmetric

Answer 1

SOLUTION

TO PROVE

$\sf{{A}^{2} \: \: is \: \: symmetric }$

if either A is symmetric or A is skew symmetric

CONCEPT TO BE IMPLEMENTED

1. A matrix A is said to be symmetric if

$\sf{{A}^{t} = A}$

2. A matrix A is said to be skew symmetric if

$\sf{{A}^{t} = - A}$

$3. \: \sf{ {(AB)}^{t} = {B}^{t}{A}^{t} }$

EVALUATION

First suppose that A is symmetric

Then

$\sf{{A}^{t} = A}$

Now

$\sf{{({A}^{2}) }^{t} }$

$= \sf{ {(AA)}^{t} }$

$= \sf{ {(A)}^{t} } \sf{ {(A)}^{t} }$

$= \sf{A.A }$

$= \sf{{A}^{2} }$

$\therefore \: \: \: \sf{{A}^{2} \: \: is \: \: symmetric }$

Next suppose that A is skew symmetric

Then

$\sf{{A}^{t} = - A}$

Now

$\sf{{({A}^{2}) }^{t} }$

$= \sf{ {(AA)}^{t} }$

$= \sf{ {(A)}^{t} } \sf{ {(A)}^{t} }$

$= \sf{( - A).( - A) }$

$= \sf{{A}^{2} }$

$\therefore \: \: \: \sf{{A}^{2} \: \: is \: \: symmetric }$

Hence proved

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