Let π΄ be an π Γ π matrix over the set of all real numbers β. Let π΅ be aβ¦
2024
Let π΄ be an π Γ π matrix over the set of all real numbers β. Let π΅ be a matrix obtained from π΄ by swapping two rows. Which of the following statements is/are TRUE?
- A.
The determinant of π΅ is the negative of the determinant of π΄
- B.
If π΄ is invertible, then π΅ is also invertible
- C.
If π΄ is symmetric, then π΅ is also symmetric
- D.
If the trace of π΄ is zero, then the trace of π΅ is also zero
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Correct answer: A, B
Answer: The true statements are: the determinant of B is the negative of the determinant of A, and if A is invertible then B is also invertible.
Determinant change: Swapping two rows is equivalent to left-multiplying A by a row-permutation matrix P that represents the transposition; such a P has determinant β1. Hence det(B) = det(P)Β·det(A) = βdet(A).
Invertibility: If det(A) β 0 then det(B) = βdet(A) β 0, so B is invertible. Equivalently, B = PΒ·A and P is invertible, so B is invertible whenever A is.
Symmetry: Not necessarily preserved. Example: A = [[1,0],[0,2]] is symmetric, but swapping the two rows gives B = [[0,2],[1,0]], which is not symmetric.
Trace: Not preserved in general. Using the same example, trace(A) = 1+2 = 3 while trace(B) = 0+0 = 0, so the trace can change after swapping rows.
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