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Question:

Using properties of determinants, prove that: \begin{vmatrix} 1+a & 1 & 1 \ 1 & 1+b & 1 \ 1 & 1 & 1+c \end{vmatrix} = abc+bc+ca+ab

Solution:

LHS=\begin{vmatrix} 1+a & 1 & 1 \ 1 & 1+b & 1 \ 1 & 1 & 1+c \end{vmatrix}
Applying R1 \to R1 - R2, R2 \to R2 - R3 and row column transformation,
=\begin{vmatrix} a & -b & 0 \ 0 & b & -c \ 1 & 1 & 1+c \end{vmatrix}
=a(b(1+c)+c)+1(bc)=ab+abc+ac+bc=abc+ab+bc+ca=RHS
Hence proved.