If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $A_2=\left[\begin{array}{ll}\alpha &…

If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $A_2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]$, then
  1. $\alpha=2 a b, \beta=a^2+b^2$
  2. $\alpha=\mathrm{a}_2+\mathrm{b}_2, \beta=\mathrm{ab}$
  3. $\alpha=a^2+b^2, \beta=2 a b$
  4. $\alpha=a^2+b^2, \beta=a^2-b^2$

Solution

$A^2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ $\alpha=a^2+b^2 ; \beta=2 a b$

Asked in: JEE Main 2003

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