Find the determinant of \begin{bmatrix}2 & 1 & 3\0 & 4 & -1\5 & 0 & 2\end{\bmatrix}.

Step-by-step solution

1. Expand along the first row: \det=2cdotdet\begin{bmatrix}4&-1\0&2\end{\bmatrix}-1cdotdet\begin{bmatrix}0&-1\5&2\end{\bmatrix}+3cdotdet\begin{bmatrix}0&4\5&0\end{\bmatrix}
2. Compute the 2×2 determinants: \det\begin{bmatrix}4&-1\0&2\end{\bmatrix}=8, \det\begin{bmatrix}0&-1\5&2\end{\bmatrix}=5, \det\begin{bmatrix}0&4\5&0\end{\bmatrix}=-20
3. Substitute: $\det =2\times 8-1\times 5+3\times (-20)=16-5-60$
4. Simplify: $16-5-60=-49$

Concept tested: Cofactor expansion for 3×3 determinants

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