Question

If $A$ and $B$ are invertible square matrices of order $n$, then which of the following isnot correct?

• A. $\det(AB) = \det(A) \cdot \det(B)$
• B. $\det(kA) = k^n \det(A)$
• C. $\det(A + B) = \det(A) + \det(B)$
• D. $\det(A^{-1}) = \frac{1}{\det(A)}$

Hint:

A common pitfall is assuming that operations like determinant or trace are fully linear. Remember: Determinants distribute over multiplication but NOT over addition. (Conversely, the Trace operator distributes over addition but not multiplication).

Answer 1

The Correct Option is C

To solve this problem, we need to determine which statement about determinants is incorrect. Let's analyze each option one by one:

1. \(\det(AB) = \det(A) \cdot \det(B)\)
2. \(\det(kA) = k^n \det(A)\)
3. \(\det(A + B) = \det(A) + \det(B)\)
4. \(\det(A^{-1}) = \frac{1}{\det(A)}\)

Conclusion: The statement that is not correct is \(\det(A + B) = \det(A) + \det(B)\). Determinants are not linear functions, so this property does not hold true.