Consider the following two statements with respect to the matrices $A_{m \times n}$, $B_{n \times m}$, $C_{n \times n}$ and $D_{n \times n}$.
Statement 1: $tr(AB) = tr(BA)$
Statement 2: $tr(CD) = tr(DC)$
where $tr()$ represents the trace of a matrix. Which one of the following holds?
Statement 1: $tr(AB) = tr(BA)$
Statement 2: $tr(CD) = tr(DC)$
where $tr()$ represents the trace of a matrix. Which one of the following holds?
Show Hint
- Statement 1 is correct and Statement 2 is wrong.
- Statement 1 is wrong and Statement 2 is correct.
- Both Statement 1 and Statement 2 are correct.
- Both Statement 1 and Statement 2 are wrong.
The Correct Option is C
Solution and Explanation
The question involves understanding properties of the trace function of matrices. Let's address each statement individually to determine their correctness.
- Statement 1: \(tr(AB) = tr(BA)\)
The trace of a matrix is defined as the sum of its diagonal elements. One of the important properties of the trace function is that for two matrices \( A \) and \( B \), where \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times m \) matrix, the trace of the product of these two matrices is equal irrespective of their order.
This means that \(\text{tr}(AB) = \text{tr}(BA)\). This is a known property that results from the cyclic property of the trace function in linear algebra.
| Step | Reason |
|---|---|
| Cyclic property | \(\text{tr}(XY) = \text{tr}(YX)\) for matrices of compatible dimensions. |
Thus, Statement 1 is correct.
- Statement 2: \(tr(CD) = tr(DC)\)
Similarly to the first statement, let's analyze this. Matrices \( C \) and \( D \) are both \( n \times n \) matrices. Applying the cyclic property of the trace again in this context, we find:
Since \( C \) and \( D \) are square matrices of the same order, the trace property holds: \(\text{tr}(CD) = \text{tr}(DC)\). The cyclic nature of the trace gives us this equality, regardless of matrix size, provided dimensions are compatible.
Thus, Statement 2 is also, indeed, correct.
Therefore, both statements given in the question are correct.
Conclusion: The correct answer is "Both Statement 1 and Statement 2 are correct."
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