Find the determinant of the matrix $A=\left(\begin{array}{ccc}2& -1& 3\\ 0& 4& 5\\ 1& 2& 0\end{array}\right)$.

Step-by-step solution

1. Write the determinant using cofactor expansion along the first row.
2. Compute the first term: $2\times \det \left(\begin{array}{cc}4& 5\\ 2& 0\end{array}\right)=2\times (4\cdot 0-5\cdot 2)=2\times (-10)=-20$.
3. Compute the second term: $-(-1)\times \det \left(\begin{array}{cc}0& 5\\ 1& 0\end{array}\right)=+1\times (0\cdot 0-5\cdot 1)=-5$.
4. Compute the third term: $3\times \det \left(\begin{array}{cc}0& 4\\ 1& 2\end{array}\right)=3\times (0\cdot 2-4\cdot 1)=-12$.
5. Add the three contributions: $-20-5-12=-37$.

Concept tested: Determinant of a 3×3 matrix

Published 12 September 2026 · How this solution was prepared and checked