First slide

Introduction to Determinants

Question

The value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ b + c & c + a & a + b \\ b + c - a & c + a - b & a + b - c \end{matrix} |$ is

Moderate

A

$a b c$
B

$a + b + c$
C

$a b + b c + c a$
D

None of these

Solution

$| \begin{matrix} 1 & 1 & 1 \\ b + c & c + a & a + b \\ b + c - a & c + a - b & a + b - c \end{matrix} |$
$\begin{matrix} R_{2} \to 2 R_{2} - R_{3} \end{matrix}$
$| \begin{matrix} 1 & 1 & 1 \\ a + b + c & a + b + c & a + b + c \\ b + c - a & c + a - b & a + b - c \end{matrix} |$
$(a + b + c) | \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ b + c - a & c + a - b & a + b - c \end{matrix} | = 0$

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