if A is a square matrix ,expressed as A= x+y ,where x is a symmetric matrix and y is skew symmetric , find value of x and y ​

Question

if A is a square matrix ,expressed as A= x+y ,where x is a symmetric matrix and y is skew symmetric , find value of x and y

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Elliana 1 month 2021-09-14T22:27:13+00:00 1 Answer 0 views 0

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    2021-09-14T22:28:34+00:00

    SOLUTION

    GIVEN

    A is a square matrix expressed as A= X + Y ,

    where X is a symmetric matrix and Y is skew symmetric matrix

    TO DETERMINE

    The value of X and Y

    CONCEPT TO BE IMPLEMENTED

    1. A matrix M is said to be symmetric matrix if

     \sf{{M}^{t}  =M }

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

     \sf{{M}^{t}  = - M }

    EVALUATION

    Here it is given that

    A= X + Y ………… (1)

    Here it is also stated that X is a symmetric matrix and Y is skew symmetric matrix

    So

     \sf{{X}^{t}  = X}

     \sf{{Y}^{t}  =  - Y}

    Taking transpose in both sides of Equation (1) we get

     \sf{{A}^{t}  = {(X + Y)}^{t} }

     \implies \sf{{A}^{t}  = {X}^{t}  + {Y}^{t} }

     \implies \sf{{A}^{t}  = {X} -  {Y}} \:  \:  \: .......(2)

    Solving Equation (1) & Equation (2) we get

     \displaystyle \sf{X =  \frac{1}{2} \big( A +  {A}^{t} \big) }

     \displaystyle \sf{Y =  \frac{1}{2} \big( A  -   {A}^{t} \big) }

    Which is the required value of X and Y

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