First slide

Introduction to Determinants

Question

If a, b, c, d, e and f are in G.P., then the value of $|\begin{array}{lll} a^{2} d^{2} x \\ b^{2} e^{2} y \\ c^{2} f^{2} z \end{array}|$ depends on

Moderate

A

x and y
B

x and z
C

y and z
D

independent of x, y and z

Solution

Since a, b, c, d, e, f are in G. P. and if r is the common ratio of the G.P., then
$\begin{matrix} b = ar \\ c = {ar}^{2} \\ d = {ar}^{3} \\ e = {ar}^{4} \\ f = {ar}^{5} \end{matrix}$
Therefore, given determinant is
$\begin{matrix} |\begin{array}{ccc} a^{2} & a^{2} r^{6} & x \\ a^{2} r^{2} & a^{2} r^{8} & y \\ a^{2} r^{4} & a^{2} r^{10} & z \end{array}| = a^{2} a^{2} r^{6} |\begin{array}{ccc} 1 & 1 & x \\ r^{2} & r^{2} & y \\ r^{4} & r^{4} & z \end{array}| \\ = a^{4} r^{6} (0) = 0 [∵ C_{1}, C_{2} are identical] \end{matrix}$

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