Find the determinant of the matrix \begin{bmatrix}6 & 2 & 1\\0 & -3 & 4\\5 & 7 & -2\end{bmatrix}.

Step-by-step solution

1. Write the determinant using cofactor expansion along the first row.
2. Compute the first term: $6\times \det \left[\begin{array}{cc}-3& 4\\ 7& -2\end{array}\right]=6\times ((-3)(-2)-4\cdot 7)=6\times (6-28)=-132$.
3. Compute the second term with sign $-$: $-2\times \det \left[\begin{array}{cc}0& 4\\ 5& -2\end{array}\right]=-2\times (0\cdot (-2)-4\cdot 5)=-2\times (-20)=40$.
4. Compute the third term: $+1\times \det \left[\begin{array}{cc}0& -3\\ 5& 7\end{array}\right]=1\times (0\cdot 7-(-3)\cdot 5)=15$.
5. Add the three contributions: $-132+40+15=-77$.

Concept tested: Determinant of 3×3 matrix by cofactor expansion

Published 19 September 2026 · How this solution was prepared and checked