$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is its transpose. The sum and differences of these matrices and defined as $$\left[ S \right] = \left[ A \right] + {\left[ A \right]^T}$$ and $$\left[ D \right] = \left[ A \right] - {\left[ A \right]^T}$$ respectively. Which of the following statements is true?

Consider a finite sequence of random values $$X = \left\{ {{x_1},{x_2},{x_3}, - - - - - {x_n}} \right\}..$$ Let $${\mu _x}$$ be the mean and $${\sigma _x}$$ be the standard deviation of $$X.$$ Let another finite sequence $$Y$$ of equal length be derived from this $${y_i} = a.{x_i} + b,$$ where $$a$$ and $$b$$ are positive constants. Let $${\mu _y}$$ be the mean and $${\sigma _y}$$ be the standard deviation of this sequence. Which one of the following statements is incorrect?

If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?

An application loads $$100$$ libraries at startup. Loading each library requires exactly one disk access. The seek time of the disk to random location is given as $$10$$ $$ms$$. Rotational speed of disk is $$6000$$ $$rpm.$$ If all $$100$$ libraries are loaded from random locations on the disk, how long does it take to load all libraries? (The time to transfer data from the disk block once the head has been positioned at the start of the block may be neglected).