A hydraulic structure has four gates which operate independently. The probability of failure of each gate is $$0.2.$$ Given that gate $$1$$ has failed, the probability that both gates $$2$$ and $$3$$ will fail is

Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\,\,{\left[ D \right]_{5x3,}}$$ $$\,\,{\left[ E \right]_{5x5,}}$$ $$\,\,{\left[ F \right]_{5x1,}}$$ are given. Matrices $$\left[ B \right]$$ and $$\left[ E \right]$$ are symmetric. Following statements are made with respect to their matrices.
$$(I)$$ Matrix product $$\,\,{\left[ F \right]^T}\,\,$$ $$\,\,{\left[ C \right]^T}\,\,$$ $$\,\,\left[ B \right]\,\,$$ $$\,\,\left[ C \right]\,\,$$ $$\,\,\left[ F \right]\,\,$$ is a scalar.
$$(II)$$ Matrix product $$\,\,{\left[ D \right]^T}\,\,$$ $$\,\left[ F \right]\,\,$$ $$\,\left[ D \right]\,\,$$ is always symmetric.
With reference to above statements which of the following applies?

The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are

The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at