Math, asked by nehajosephbb, 9 months ago

. a) Let A = [ 2 x 4 5 3 8 4 y 9] , if A is symmetric , find value of x and y b) From part (a) verify that AA' and A + A' are symmetric matrices [ A' means AT ]

Answers

Answered by MaheswariS
2

\textbf{Given:}

A=\left(\begin{array}{ccc}2&x&4\\5&3&8\\4&y&9\end{array}\right)

\textbf{To find:}

\text{The values of x and y}

\textbf{Solution:}

(a)

\text{Since A is symmetric matrix, we have}

A=A'

\left(\begin{array}{ccc}2&x&4\\5&3&8\\4&y&9\end{array}\right)=\left(\begin{array}{ccc}2&5&4\\x&3&y\\4&8&9\end{array}\right)

\text{Equating the corresponding elements, we get}

x=5\;\text{and}\;y=8

(b)

AA'

=\left(\begin{array}{ccc}2&5&4\\5&3&8\\4&8&9\end{array}\right)\left(\begin{array}{ccc}2&5&4\\5&3&8\\4&8&9\end{array}\right)

=\left(\begin{array}{ccc}4+25+16&10+15+32&8+40+36\\10+15+32&25+9+64&20+24+72\\8+40+36&20+24+72&16+64+81\end{array}\right)

\implies\bf\,AA'=\left(\begin{array}{ccc}45&57&84\\57&98&116\\84&116&161\end{array}\right)

\text{It is clear that, $\bf\,AA'=(AA')'$}

\text{Hence $AA'$ is a symmetric matrix}

A+A'=\left(\begin{array}{ccc}2&5&4\\5&3&8\\4&8&9\end{array}\right)+\left(\begin{array}{ccc}2&5&4\\5&3&8\\4&8&9\end{array}\right)

\implies\bf\,A+A'=\left(\begin{array}{ccc}4&10&8\\10&6&16\\8&16&18\end{array}\right)

\text{It is clear that, $\bf\,A+A'=(A+A')'$}

\textbf{Hence $A+A'$ is a symmetric matrix}

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