Medium Practice — Set 2

Quick Recap — Determinant Techniques

• Properties: $|kA|=k^n|A|$, $|A^{-1}|=\tfrac1{|A|}$, $|\operatorname{adj}A|=|A|^{n-1}$, $A(\operatorname{adj}A)=|A|I$.
• Row/column ops simplify evaluation; proportional rows/columns give $0$.
• Systems: homogeneous $AX=0$ has non-trivial solutions iff $|A|=0$; Cramer needs $|A|\ne0$.
• Odd-order skew-symmetric determinant $=0$.