Let $A$ be the $3\times 3$ matrix with entries $a_{ij}=i+j$ ($i,j=1,2,3$). Compute $\det (A)$.

Step-by-step solution

1. Write the matrix: A=\begin{bmatrix}2&3&4\3&4&5\4&5&6\end{\bmatrix}
2. Subtract the first row from the second: new second row = \begin{bmatrix}1&1&1\end{\bmatrix}
3. Subtract the new second row from the third: new third row = \begin{bmatrix}1&1&1\end{\bmatrix}
4. Now rows 2 and 3 are identical, hence the matrix has two equal rows
5. A matrix with two equal rows has determinant zero
6. Thus $det(A)=0$

Concept tested: Row operations to detect linear dependence

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