Question

(b) Show that \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \] :

Hint:

For determinant proofs, use row and column operations to simplify the matrix.

Answer 1

Expand the determinant using row or column operations. First, subtract the first column from the second and third columns: \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} \rightarrow \begin{vmatrix} 1 + a & b & c \\ 1 & b & c \\ 1 & b & c \end{vmatrix}. \] Using the properties of determinants and simplifying yields: \[ abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right). \]