If the matrix `A = [[lambda_(1)^(2), lambda_(1)lambda_(2), lambda_(1) lambda_(3)],[lambda_(2)lambda_(1),lambda_(2)^(2),lambda_(2)lambda_(3)],[lambda_(3)lambda_(1),lambda_(3)lambda_(2),lambda_(3)^(2)]]` is idempotent,
the value of `lambda_(1)^(2) + lambda_(2)^(2) + lambda _(3)^(2)` is

Answer

1

Explanation

`because A^(2) = Acdot A = [[lambda_(1)^(2), lambda_(1) lambda_(2), lambda_(1) lambda_(3) ],[lambda_(2)lambda_(1), lambda_(2)^(2),lambda_(2)lambda_(3)],[lambda_(3)lambda_(1),lambda_(3)lambda_(2),lambda_(3)^(2)]] [[lambda_(1)^(2), lambda_(1) lambda_(2), lambda_(1) lambda_(3) ],[lambda_(2)lambda_(1), lambda_(2)^(2),lambda_(2)lambda_(3)],[lambda_(3)lambda_(1),lambda_(3)lambda_(2),lambda_(3)^(2)]]`
` = [[lambda_(1)^(2)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)), lambda_(1) lambda_(2)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)), lambda_(1) lambda_(3)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)) ],[lambda_(2)lambda_(1)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)), lambda_(2)^(2) (lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)),lambda_(2)lambda_(3)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2))],[lambda_(3)lambda_(1)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)),lambda_(3)lambda_(2)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2)),lambda_(3)^(2)(lambda_(1)^(2)+lambda_(2)^(2)+lambda_(3)^(2))]]`
`= (lambda_(1)^(2) + lambda_(2)^(2) + lambda_(3)^(2) ) A`
Given, A is idempotent
`rArr A^(2) = A`
`= lambda_(1)^(2) + lambda_(2)^(2) + lambda_(3)^(2) = 1`