First slide

Introduction to Determinants

Question

If $x_{1}, x_{2}, x_{3}, \dots, x_{13}$ are in A.P. then the value of $|\begin{array}{ccc} e^{x_{1}} & e^{x_{4}} & e^{x_{7}} \\ e^{x_{4}} & e^{x_{7}} & e^{x_{10}} \\ e^{x_{7}} & e^{x_{10}} & e^{x_{13}} \end{array}|$ is

Moderate

A

27
B

0
C

1
D

9

Solution

Taking e^{x 1} common from R₁, e^x4 common from R₂and e^x7 from R_3, we get
$Δ = e^{x_{1} + x_{4} + x_{7}} Δ_{1}$
where $Δ_{1} = |\begin{array}{ccc} 1 & e^{3 d} & e^{6 d} \\ 1 & e^{3 d} & e^{6 d} \\ 1 & e^{3 d} & e^{6 d} \end{array}| = 0$
where d is common difference of A.P.
$∴ Δ = 0$

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