Question

Evaluate the determinant \[ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}. \]

Hint:

Use row operations to simplify determinants before expansion; factor and expand carefully.

Answer 1

We expand the determinant: \[ D = \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}. \] Subtract the first row from the second and third rows to simplify: \[ R_2 \to R_2 - R_1, \quad R_3 \to R_3 - R_1, \] giving \[ D = \begin{vmatrix} 1+a & 1 & 1 \\ 1 - (1+a) & (1+b) - 1 & 1 - 1 \\ 1 - (1+a) & 1 - 1 & (1+c) - 1 \end{vmatrix} = \begin{vmatrix} 1+a & 1 & 1 \\ - a & b & 0 \\ - a & 0 & c \end{vmatrix}. \] Now expand along the first row: \[ D = (1+a) \begin{vmatrix} b & 0 \\ 0 & c \end{vmatrix} - 1 \begin{vmatrix} -a & 0 \\ -a & c \end{vmatrix} + 1 \begin{vmatrix} -a & b \\ -a & 0 \end{vmatrix}. \] Calculate each minor: \[ = (1+a)(b \cdot c - 0) - 1(-a \cdot c - 0) + 1(-a \cdot 0 - (-a) b) \] \[ = (1+a) b c + a c + a b. \] Therefore, \[ \boxed{ D = b c (1+a) + a c + a b = abc + bc + ac + ab. } \]