JEE MAIN - Mathematics (2003 - No. 15)
If $$A = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$ and $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta & \alpha \cr
} } \right]$$, then
$$\alpha = 2ab,\,\beta = {a^2} + {b^2}$$
$$\alpha = {a^2} + {b^2},\,\beta = ab$$
$$\alpha = {a^2} + {b^2},\,\beta = 2ab$$
$$\alpha = {a^2} + {b^2},\,\beta = {a^2} - {b^2}$$
Explanation
$${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta & \alpha \cr
} } \right] = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]\left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$
$$ = \left[ {\matrix{ {{a^2} + {b^2}} & {2ab} \cr {2ab} & {{a^2} + {b^2}} \cr } } \right]$$
$$\alpha = {a^2} + {b^2};\,\,\beta = 2ab$$
$$ = \left[ {\matrix{ {{a^2} + {b^2}} & {2ab} \cr {2ab} & {{a^2} + {b^2}} \cr } } \right]$$
$$\alpha = {a^2} + {b^2};\,\,\beta = 2ab$$
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