From the list below, select all TRUE properties of determinants. (But do not select any false statements!) Assume that all matrices are square n x n matrices, and In represents the identity 1 X n matrix. Odet(-In) = -1 for any matrix size n. Odet (AB) = det(A) det (B) det (PAP-¹) = det (A), assuming Pis invertible. Odet (In) = 1 for any matrix size n. Odet(A + B) = det(A) + det(B) Odet (CA) = c. det(A), where c is a scalar (real number). Of UTU = In. then det (U) = 1. det (AB-¹) Odet (Ak) = (det (A))k = det (A) det (B) 1 assuming det (B) ‡ 0. where k is a positive integer.
From the list below, select all TRUE properties of determinants. (But do not select any false statements!) Assume that all matrices are square n x n matrices, and In represents the identity 1 X n matrix. Odet(-In) = -1 for any matrix size n. Odet (AB) = det(A) det (B) det (PAP-¹) = det (A), assuming Pis invertible. Odet (In) = 1 for any matrix size n. Odet(A + B) = det(A) + det(B) Odet (CA) = c. det(A), where c is a scalar (real number). Of UTU = In. then det (U) = 1. det (AB-¹) Odet (Ak) = (det (A))k = det (A) det (B) 1 assuming det (B) ‡ 0. where k is a positive integer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:From the list below, select all **TRUE** properties of determinants. (But do not select any false statements!)
Assume that all matrices are square \( n \times n \) matrices, and \( I_n \) represents the identity \( n \times n \) matrix.
- \(\det(-I_n) = -1\) for any matrix size \( n \).
- \(\det(AB) = \det(A) \cdot \det(B)\)
- \(\det(PAP^{-1}) = \det(A)\), assuming \( P \) is invertible.
- \(\det(I_n) = 1\) for any matrix size \( n \).
- \(\det(A + B) = \det(A) + \det(B)\)
- \(\det(cA) = c \cdot \det(A)\), where \( c \) is a scalar (real number).
- If \( U^TU = I_n \), then \(\det(U) = 1\).
- \(\det\left(AB^{-1}\right) = \frac{\det(A)}{\det(B)}\), assuming \(\det(B) \neq 0\).
- \(\det(A^k) = (\det(A))^k\) where \( k \) is a positive integer.
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