The value of $\left| {\,\begin{array}{*{20}{c}}{{5^2}}&{{5^3}}&{{5^4}}\\{{5^3}}&{{5^4}}&{{5^5}}\\{{5^4}}&{{5^5}}&{{5^7}}\end{array}\,} \right|$ is

Solve the equations $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0, a \neq 0$

If $a, b, c > 0 \, and \, x, y, z \in R$ , then the determinant $\left|{\begin{array}{*{20}{c}}{{{\left( {{a^x}\, + \,\,{a^{ – x}}} \right)}^2}}&{{{\left( {{a^x}\, – \,\,{a^{ – x}}} \right)}^2}}&1\\{{{\left( {{b^y}\, + \,\,{b^{ – y}}} \right)}^2}}&{{{\left( {{b^y}\, – \,\,{b^{ – y}}} \right)}^2}}&1\\{{{\left( {{c^z}\, + \,\,{c^{ – z}}} \right)}^2}}&{{{\left( {{c^z}\, – \,\,{c^{ – z}}} \right)}^2}}&1\end{array}} \right|$ $=$

$\left| {\,\begin{array}{*{20}{c}}{{a^2}}&{{b^2}}&{{c^2}}\\{{{(a + 1)}^2}}&{{{(b + 1)}^2}}&{{{(c + 1)}^2}}\\{{{(a – 1)}^2}}&{{{(b – 1)}^2}}&{{{(c – 1)}^2}}\end{array}\,} \right| = $

By using properties of determinants, show that:

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