For a $3 \times 3$ matrix $A$, $\det(A^2)$ equals
A$(\det A)^2$
B$2 \det A$
C$\det A$
D$A^2$
Answer & Solution
Correct answer: A. $(\det A)^2$
1. DETERMINANT is MULTIPLICATIVE: $\det(AB) = \det(A)\det(B)$.
2. With $B = A$: $\det(A^2) = \det(A)^2$.
3. Generalisation: $\det(A^k) = (\det A)^k$.
4. Other useful facts: $\det(A^{-1}) = 1/\det A$; $\det(A^T) = \det A$; $\det(cA) = c^n \det A$ for $n \times n$ matrix.
5. Other options miss the multiplicative property.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §4.3 (Determinant properties)._
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