If  A=xyyx and  A2=abba , then  a+b=

# If  $A=\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]$ and  ${A}^{2}=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ , then  $a+b=$

1. A

${x}^{2}+{y}^{2}+xy$

2. B

${x}^{2}+{y}^{2}-xy$

3. C

${\left(x+y\right)}^{2}$

4. D

${\left(x-y\right)}^{2}$

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### Solution:

$\begin{array}{l}A=\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]\\ {A}^{2}=\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]\left[\begin{array}{cc}x& y\\ y& x\end{array}\right]=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]\\ ⇒\left[\begin{array}{cc}{x}^{2}+{y}^{2}& 2xy\\ 2xy& {x}^{2}+{y}^{2}\end{array}\right]\\ ⇒\left[\begin{array}{cc}{x}^{2}+{y}^{2}& 2xy\\ 2xy& {x}^{2}+{y}^{2}\end{array}\right]=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]\\ ⇒a={x}^{2}+{y}^{2},b=2xy\\ a+b={\left(x+y\right)}^{2}\end{array}$

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