Let $A=\left[\begin{array}{ccc}1& 0& 2\\ 3& -1& 4\\ 5& 2& -3\end{array}\right]$ and let $B$ be the matrix obtained by interchanging the first and third rows of $A$. If $\det A=7$, what is $\det B$?

Step-by-step solution

1. Recall the property: swapping two rows of a matrix multiplies its determinant by $-1$.
2. Matrix $B$ is obtained from $A$ by exactly one row interchange (rows 1 and 3).
3. Therefore $\det B=-\det A$.
4. Given $\det A=7$, compute $\det B=-7$.
5. Hence the determinant of $B$ is $-7$.

Concept tested: Effect of row interchange on determinant

Published 15 September 2026 · How this solution was prepared and checked