Let A,B,C be n×n real matrices and λ a real number. Which of the following statements is wrong (in general)?

a. $tr(A+B)=tr(A)+tr(B)$  

b. $det(λA)=λ^{n}det(A)$  

c. $det(A)=det(A^T)$  

d. $tr(λA)=λtr(A)$  

e. $tr(ABC)=tr(BAC)$

To determine which of the given statements is generally wrong for $ n \times n $ real matrices $ A, B, C $ and a real number $\lambda$, let's analyze each statement:

a. $ \text{tr}(A+B) = \text{tr}(A) + \text{tr}(B) $

This is true. The trace of a matrix is the sum of its diagonal elements, and the trace of the sum of two matrices is the sum of their traces.

b. $ \det(\lambda A) = \lambda^n \det(A) $

This is true. The determinant of a scalar multiple of a matrix is the scalar raised to the power of the matrix's order (n) multiplied by the determinant of the matrix.

c. $ \det(A) = \det(A^T) $

This is true. The determinant of a matrix is equal to the determinant of its transpose.

d. $ \text{tr}(\lambda A) = \lambda \text{tr}(A) $

This is true. The trace of a scalar multiple of a matrix is the scalar multiplied by the trace of the matrix.

e. $ \text{tr}(ABC) = \text{tr}(BAC) $

This is not generally true. The trace of the product of three matrices does not necessarily satisfy this cyclic property. While $\text{tr}(ABC) = \text{tr}(BCA) = \text{tr}(CAB)$ holds due to the cyclic property of the trace, $\text{tr}(ABC) \neq \text{tr}(BAC)$ in general.

Therefore, the statement that is wrong in general is:

**e. $ \text{tr}(ABC) = \text{tr}(BAC) $**

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What are the properties of the trace function in linear algebra?
2. How does the determinant of a matrix change under matrix transposition?
3. What is the significance of the cyclic property of the trace in matrix algebra?
4. Can you provide examples of cyclic and non-cyclic permutations of matrix products and their traces?
5. How is the determinant affected by elementary row operations?

**Tip:** When dealing with properties of matrices, always consider both the definitions and the implications of operations like addition, multiplication, and transposition. This ensures a thorough understanding of how these properties interact.

Explore the properties of matrices with a focus on trace and determinant operations. Analyze statements involving matrix traces, determinants, and transpositions. Identify which statements are generally incorrect for n×n real matrices and a real number λ.

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