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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Answer 1

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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