First slide

Introduction to Determinants

Question

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

Easy

A

1+x+y+z
B

x +y +z
C

0
D

1

Solution

$| \begin{matrix} x 1 y + z \\ y 1 z + x \\ z 1 x + y \end{matrix} |$
$C_{1} \to C_{1} + C_{2} + C_{3}$
$|\begin{matrix} 1 + x + y + z & 1 & y + z \\ 1 + x + y + z & 1 & z + x \\ 1 + x + y + z & 1 & x + y \end{matrix}|$
$\begin{matrix} = (1 + x + y + z) | \begin{matrix} 1 & 1 & y + z \\ 1 & 1 & z + x \\ 1 & 1 & x + y \end{matrix} | \\ = 0 \end{matrix}$

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