If A and B are square matrices of size $n \times n$, then which of the following statement is not true.
Show Hint
- $\det(AB) = \det(A)\det(B)$
- $\det(kA) = k^{n}\det(A)$
- $\det(A + B) = \det(A) + \det(B)$
- $\det(A^{T}) = \det(A)$
The Correct Option is C
Solution and Explanation
Step 1: Concept
Determinant properties describe how the determinant function behaves under various matrix operations such as multiplication, scaling, and transposition.
Step 2: Meaning
The determinant is a multiplicative function, meaning the determinant of a product is the product of the determinants. However, it is not a linear function with respect to matrix addition.
Step 3: Analysis
While \(\det(kA)=k^n\det(A)\) and \(\det(A^T)=\det(A)\) are standard identities, \(\det(A+B)\) generally does not equal \(\det(A)+\det(B)\). For example, if \[ A=\begin{pmatrix} 1 & amp; 0\\ 0 & amp; 0 \end{pmatrix} \quad \text{and} \quad B=\begin{pmatrix} 0 & amp; 0\\ 0 & amp; 1 \end{pmatrix}, \] then \(\det(A)=0\), \(\det(B)=0\), but \[ A+B=\begin{pmatrix} 1 & amp; 0\\ 0 & amp; 1 \end{pmatrix}, \] so \(\det(A+B)=1\).
Step 4: Conclusion
Therefore, statement (C) is the incorrect property among the given choices.
Final Answer: (C)
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