Let \(\displaystyle\alpha={a}+{b}{i}\ \text{ and }\ \beta={c}+{d}{i}\) be complex scalars and let A and B be matrices with complex entries. (a) Show that \(\displaystyle\overline{{\alpha+\beta}}=\overline{{\alpha}}+\overline{{\beta}}\ \text{ and }\ \overline{{\alpha\beta}}=\overline{{\alpha}}\overline{{\beta}}\) (b) Show that the (i,j) entries of\( \overline{AB} \text{ and } \bar{A}\bar{B}\) are equal and hence that \(\displaystyle\overline{{{A}{B}}}=\overline{{{A}}}\overline{{{B}}}\)

Suppose X is a random variable such that \(E(3X-7)=8\) and \(\displaystyle{E}{\left({\frac{{{X}^{{2}}}}{{{2}}}}\right)}={19}\). What is Var\((70-2X)\)?

3) Answer the following questions considering the complex functions given below. a) Using the definition of complex derivative, evaluate f(z) expression using derivative operation based on limiting case as \(\displaystyle\lim_{{\triangle{z}\rightarrow{0}}}\) a.1 \(\displaystyle{f{{\left({z}\right)}}}={\frac{{{1}}}{{{z}^{{2}}}}},{\left({z}\ne{}{0}\right)}\)