Evaluate: $\left[\begin{array}{cc}\mathrm{a} + \mathrm{ib}& \mathrm{c} + \mathrm{id}\\ -\mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

Find the co-factor of a12 in the following:

$\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$

Using properties of determinants, prove the following:

$\left|\begin{array}{ccc}\mathrm{\alpha }& \mathrm{\beta }& \mathrm{\gamma }\\ {\mathrm{\alpha }}^2& {\mathrm{\beta }}^2& {\mathrm{\gamma }}^2\\ \mathrm{\beta } + \mathrm{\gamma } & \mathrm{\gamma } + \mathrm{\alpha } & \mathrm{\alpha } + \mathrm{\beta }\end{array}\right| = \left(\mathrm{\alpha } - \mathrm{\beta }\right) \left(\mathrm{\beta } - \mathrm{\gamma } \right) \left(\mathrm{\gamma } - \mathrm{\alpha }\right) \left(\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma }\right)$

Write the value of the determinant $\left|\begin{array}{ccc} 2 & 3 & 4 \\ 5 & 6 & 8\\ 6\mathrm{x} & 9\mathrm{x} & 12\mathrm{x}\end{array}\right|$

Using properties of determinants prove the following:

$\left|\begin{array}{ccc}\mathrm{a} & \mathrm{b}& \mathrm{c}\\ \mathrm{a} - \mathrm{b} & \mathrm{b} - \mathrm{c} & \mathrm{c} - \mathrm{a}\\ \mathrm{b} + \mathrm{c} & \mathrm{c} + \mathrm{a} & \mathrm{a} + \mathrm{b}\end{array}\right| = {\mathrm{a}}^3 + {\mathrm{b}}^3 + {\mathrm{c}}^3 - 3\mathrm{abc}$

Find the minor of the element of second row and third column ( a₂₃ ) in the following determinant:

$\left| \begin{array}{ccc} 2 & -3 & 5\\ 6& 0& 4\\ 1& 5& -7 \end{array}\right|$

Using properties of determinants show the following:

$\left|\begin{array}{ccc}{\left( \mathrm{b} + \mathrm{c} \right)}^2 & \mathrm{ab}& \mathrm{ca}\\ \mathrm{ab} & {\left( \mathrm{a} + \mathrm{c} \right)}^{2 }& \mathrm{bc}\\ \mathrm{ac} & \mathrm{bc}& {\left( \mathrm{a} + \mathrm{b} \right)}^2\end{array}\right| = 2\mathrm{abc} ( \mathrm{a} + \mathrm{b} + \mathrm{c} {)}^3$

Using properties of determinants, prove that

$\left| \begin{array}{ccc} - {\mathrm{a}}^2& \mathrm{ab} & \mathrm{ac}\\ \mathrm{ba}& -{\mathrm{b}}^2& \mathrm{bc}\\ \mathrm{ca}& \mathrm{cb}& - {\mathrm{c}}^2\end{array} \right| = 4 {\mathrm{a}}^2{\mathrm{b}}^2{\mathrm{c}}^2$

Using matrix method, solve the following system of equations:

$\frac{2}{\mathrm{x}} + \frac{3}{\mathrm{y}} + \frac{10}{\mathrm{z}} = 4, \frac{4}{\mathrm{x}} - \frac{6}{\mathrm{y}} + \frac{5}{\mathrm{z}}, \frac{6}{\mathrm{x}} + \frac{9}{\mathrm{y}} - \frac{20}{\mathrm{z}}; \mathrm{x}, \mathrm{y}, \mathrm{z} \ne 0$

If $∆ =\left| \begin{array}{ccc}5 & 3& 8 \\ 2 & 0 & 1 \\ 1 & 2& 3 \end{array}\right|$,  white the cofactor of the element a₃₂.