If $A$ is a $4\times 4$ matrix with $\det A=5$, find $\det (\mathrm{adj}A)$. (Here $\mathrm{adj}A$ denotes the adjoint of $A$.)

Step-by-step solution

1. Recall the formula for the determinant of the adjoint: $\det (\mathrm{adj}A)=(\det A{)}^{n-1}$, where $n$ is the order of $A$.
2. Identify $n=4$ and $\det A=5$.
3. Compute $(\det A{)}^{n-1}=5^{4-1}=5^3$.
4. Calculate $5^3=125$.
5. Hence $\det (\mathrm{adj}A)=125$.

Concept tested: Determinant of the adjoint matrix

Published 13 September 2026 · How this solution was prepared and checked