If $\mathrm{a, b, c}$ are positive and unequal, show that value of the determinant $\Delta=\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is negative.

If ${D_r} = \left| {\begin{array}{*{20}{c}}{{2^{r – 1}}}&{{{2.3}^{r – 1}}}&{{{4.5}^{r – 1}}}\\x&y&z\\{{2^n} – 1}&{{3^n} – 1}&{{5^n} – 1}\end{array}} \right|$, then the value of $\sum\limits_{r = 1}^n {{D_r} = } $

Let $A$ be a $3 \times 3$ matrix with $\operatorname{det}( A )=4$. Let $R _{ i }$ denote the $i ^{\text {th }}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R _{2} \rightarrow 2 R _{2}+5 R _{3}$ on $2 A ,$ then $\operatorname{det}( B )$ is equal to …… .

The determinant $\left| {\,\begin{array}{*{20}{c}}a&b&{a\alpha + b}\\b&c&{b\alpha + c}\\{a\alpha + b}&{b\alpha + c}&0\end{array}\,} \right| = 0$, if $a,b,c$ are in

If $a,b,c$ are different and $\left| {\,\begin{array}{*{20}{c}}a&{{a^2}}&{{a^3} – 1}\\b&{{b^2}}&{{b^3} – 1}\\c&{{c^2}}&{{c^3} – 1}\end{array}\,} \right| = 0$, then