First slide

Introduction to Determinants

Question

$|\begin{matrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{matrix}| =$

Easy

A

$x y z (1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z})$
B

$x y z$
C

$1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$
D

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

Solution

By expanding, we get
$(1 + x) [(1 + y) (1 + z) - 1] - 1 (z) + 1 (- y) = x y z (1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z})$
Trick : Put $x = 1, y = 2$ and $z = 3,$ then
$|\begin{matrix} 2 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 4 \end{matrix}| = 2 (11) - 1 (3) + 1 (1 - 3) = 17$
Option (1) gives, $1 \times 2 \times 3 (1 + \frac{1}{1} + \frac{1}{2} + \frac{1}{3}) = 17$

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