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3 and 4 .Determinants and Matrices
normal
If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{d^2}}&x \\
{{b^2}}&{{e^2}}&y \\
{{c^2}}&{{f^2}}&z
\end{array}} \right|$ depends on
A
$x, y$
B
$x, z$
C
$y, z$
D
None
Solution
$\mathrm{b}=\mathrm{ar}, \mathrm{c}=\mathrm{ar}^{2}, \mathrm{d}=\mathrm{ar}^{3}, \mathrm{e}=\mathrm{ar}^{4}, \mathrm{f}=\mathrm{ar}^{6}$
$\therefore\left|\begin{array}{ccc}{a^{2}} & {a^{2} r^{6}} & {x} \\ {a^{2} r^{2}} & {a^{2} r^{3}} & {y} \\ {a^{2} r^{6}} & {a^{2} r^{10}} & {z}\end{array}\right|$
$=a^{2} \times a^{2} r^{6}\left|\begin{array}{ccc}{1} & {1} & {x} \\ {r^{2}} & {r^{2}} & {y} \\ {r^{4}} & {r^{4}} & {z}\end{array}\right|=0$
Standard 12
Mathematics
