Operating R₁→R₁-R₂, R₂→R₂-R₃

→

→

Taking (a-b) and (b-c) from R₁, R₂

→

Method 1:

For the two determinants to be equal, their difference must be 0.

Since 2 columns have only 0 as their elements, by properties of determinants

=0

Method 2:

Expanding both with C₁

LHS

=(a-b)(b-c)(-a+c)

RHS

=(a-b)(b-c)(b+c-a-b)

=(a-b)(b-c)(-a+c)

∴ LHS = RHS