Compute the determinant of x 1 1 x 1 1 xend and evaluate it for x=3.

Compute the determinant of $\begin{bmatrix}x & 1 & 1\1 & x & 1\1 & 1 & xend{\bmatrix}$ and evaluate it for $x = 3$ .

hard 4 marks Class 12 · Mathematics · Determinants

Answer

20

The correct answer is $20$ — Substitute $x = 3$ : $det = (3 - 1)^{2} (3 + 2) = 4 \times 5 = 20$

Notice the matrix has the form $x I + (J - I)$ where $J$ is the $3 \times 3$ matrix of all ones
Use known eigenvalues: For $J$ , eigenvalues are $3$ (once) and $0$ (twice); for $I$ , eigenvalues are $1$
Hence eigenvalues of the matrix are $(x - 1)$ twice and $(x + 2)$ once
Determinant is product of eigenvalues: $d e t = (x - 1)^{2} (x + 2)$
Substitute $x = 3$ : $det = (3 - 1)^{2} (3 + 2) = 4 \times 5 = 20$

Common mistake: Treating all entries as independent instead of using eigenvalue method

Concept tested: Determinant of a matrix with constant off-diagonal elements

Published 5 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked