Hard Practice — Advanced Set

Quick Recap — Matrices & Determinants (Hard)

Chain the identities: $|kA|=k^n|A|$, $|\operatorname{adj}A|=|A|^{n-1}$, $|\operatorname{adj}(\operatorname{adj}A)|=|A|^{(n-1)^2}$, $A\operatorname{adj}A=|A|I$, $|A^mB|=|A|^m|B|$. Determinants also encode area/collinearity/line equations (cross-topic with coordinate geometry), and $\begin{vmatrix}x&1&1\\1&x&1\\1&1&x\end{vmatrix}=(x-1)^2(x+2)$.