Find minor and cofactor of elements of the determinant.

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

 

Find the value of $\mathrm{x}$ if 

$\left|\begin{array}{ccc}1& 4& 20\\ 1& -2& -5\\ 1& 2\mathrm{x}& 5{\mathrm{x}}^2\end{array}\right|=0$

Find the value of $\mathrm{x}$ if

$\left|\begin{array}{ccc}1& 2\mathrm{x}& 4\mathrm{x}\\ 1& 4& 16\\ 1& 1& 1\end{array}\right|=0$

By using properties of determinant prove that

 $\left|\begin{array}{ccc}\mathrm{x}+\mathrm{y}& \mathrm{y}+\mathrm{z}& \mathrm{z}+\mathrm{x}\\ \mathrm{z}& \mathrm{x}& \mathrm{y}\\ 1& 1& 1\end{array}\right|=0$

Without expanding determinant show that

$\left|\begin{array}{ccc}\mathrm{b}+\mathrm{c}& \mathrm{bc}& {\mathrm{b}}^2{\mathrm{c}}^2\\ \mathrm{c}+\mathrm{a}& \mathrm{ca}& {\mathrm{c}}^2{\mathrm{a}}^2\\ \mathrm{a}+\mathrm{b}& \mathrm{ab}& {\mathrm{a}}^2{\mathrm{b}}^2\end{array}\right|=0$

Without expanding determinant show that

$\left|\begin{array}{ccc}\mathrm{xa}& \mathrm{yb}& \mathrm{zc}\\ {\mathrm{a}}^2& {\mathrm{b}}^2& {\mathrm{c}}^2\\ 1& 1& 1\end{array}\right|=\left|\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{bc}& \mathrm{ca}& \mathrm{ab}\end{array}\right|$

Without expanding determinant show that

$\left|\begin{array}{ccc}\mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{e}& \mathrm{d}& \mathrm{f}\\ \mathrm{u}& \mathrm{v}& \mathrm{w}\end{array}\right|=\left|\begin{array}{ccc}\mathrm{n}& \mathrm{f}& \mathrm{w}\\ \mathrm{l}& \mathrm{e}& \mathrm{u}\\ \mathrm{m}& \mathrm{d}& \mathrm{v}\end{array}\right|$

Without expanding determinant show that

$\left|\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right|=0$

If$\left|\begin{array}{ccc}\mathrm{a}& 1& 1\\ 1& \mathrm{b}& 1\\ 1& 1& c\end{array}\right|=0$ then show that $\frac{1}{1-\mathrm{a}}+\frac{1}{1-\mathrm{b}}+\frac{1}{1-\mathrm{c}}=1$