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

Step-by-step solution

1. Write the determinant using cofactor expansion along the first row.
2. Compute the first cofactor: $4\times \det \left[\begin{array}{cc}-3& 5\\ 0& 2\end{array}\right]=4((-3)\cdot 2-5\cdot 0)=4(-6)=-24$.
3. Compute the second cofactor with sign $-$: $-2\times \det \left[\begin{array}{cc}0& 5\\ 7& 2\end{array}\right]=-2(0\cdot 2-5\cdot 7)=-2(-35)=70$.
4. Compute the third cofactor: $+1\times \det \left[\begin{array}{cc}0& -3\\ 7& 0\end{array}\right]=1(0\cdot 0-(-3)\cdot 7)=21$.
5. Add the three contributions: $-24+70+21=67$.

Concept tested: Determinant of 3×3 matrix

Published 17 September 2026 · How this solution was prepared and checked