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Mark each statement T if the statement is always true or F if it's ever false. All A, B, C, and D are matrices, and I is the identity matrix. Do not assume anything beyond what is explicitly stated. (a) If A and B are m × n matrices, then both products AB™ and A™B are defined. (b) If C = D, then BC = BD. %3D (c) If BC = BD, then C = D. (d) Every square matrix is the product of elementary matrices. (e) If AB = I, then A is invertible. (f) For any A, det(-A) = – det A. (g) If A and B are square matrices with det A = 2 and det B = 3, then det(A + B) = 5. (h) If two rows of a square matrix A are the same, then det A = 0. (i) If det A = 5, then A is invertible and det(A¯') =. (j) If A² = 0, then A is singular.

Mark each statement T if the statement is always true or F if it's ever false. All A, B, C, and D are matrices, and I is the identity matrix. Do not assume anything beyond what is explicitly stated. (a) If A and B are m × n matrices, then both products AB™ and A™B are defined. (b) If C = D, then BC = BD. %3D (c) If BC = BD, then C = D. (d) Every square matrix is the product of elementary matrices. (e) If AB = I, then A is invertible. (f) For any A, det(-A) = – det A. (g) If A and B are square matrices with det A = 2 and det B = 3, then det(A + B) = 5. (h) If two rows of a square matrix A are the same, then det A = 0. (i) If det A = 5, then A is invertible and det(A¯') =. (j) If A² = 0, then A is singular.

Advanced Engineering Mathematics
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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# Matrix Theory: True or False Statements **Instructions:** Mark each statement T if the statement is always true or F if it’s ever false. All \( A \), \( B \), \( C \), and \( D \) are matrices, and \( I \) is the identity matrix. Do not assume anything beyond what is explicitly stated. **Statements:** (a) If \( A \) and \( B \) are \( m \times n \) matrices, then both products \( AB^\top \) and \( A^\top B \) are defined. (b) If \( C = D \), then \( BC = BD \). (c) If \( BC = BD \), then \( C = D \). (d) Every square matrix is the product of elementary matrices. (e) If \( AB = I \), then \( A \) is invertible. (f) For any \( A \), \( \text{det} (-A) = -\text{det} A \). (g) If \( A \) and \( B \) are square matrices with \( \text{det}(A) = 2 \) and \( \text{det}(B) = 3 \), then \( \text{det}(A + B) = 5 \). (h) If two rows of a square matrix \( A \) are the same, then \( \text{det} A = 0 \). (i) If \( \text{det} A = 5 \), then \( A \) is invertible and \( \text{det}(A^{-1}) = \frac{1}{5} \). (j) If \( A^2 = 0 \), then \( A \) is singular. ### Detailed Explanations: * **Statement (a):** - **Analysis:** For the products \( AB^\top \) and \( A^\top B \) to be defined, the inner dimensions of the resulting matrices must match. Both matrices \( A \) and \( B \) are \( m \times n \), so \( B^\top \) (the transpose of \( B \)) would be \( n \times m \). Thus, \( AB^\top \) would result in an \( m \times m \) matrix, and \( A^\top \) would be \( n \times m \) while \( B \) remains \( m \times n
Transcribed Image Text:# Matrix Theory: True or False Statements **Instructions:** Mark each statement T if the statement is always true or F if it’s ever false. All \( A \), \( B \), \( C \), and \( D \) are matrices, and \( I \) is the identity matrix. Do not assume anything beyond what is explicitly stated. **Statements:** (a) If \( A \) and \( B \) are \( m \times n \) matrices, then both products \( AB^\top \) and \( A^\top B \) are defined. (b) If \( C = D \), then \( BC = BD \). (c) If \( BC = BD \), then \( C = D \). (d) Every square matrix is the product of elementary matrices. (e) If \( AB = I \), then \( A \) is invertible. (f) For any \( A \), \( \text{det} (-A) = -\text{det} A \). (g) If \( A \) and \( B \) are square matrices with \( \text{det}(A) = 2 \) and \( \text{det}(B) = 3 \), then \( \text{det}(A + B) = 5 \). (h) If two rows of a square matrix \( A \) are the same, then \( \text{det} A = 0 \). (i) If \( \text{det} A = 5 \), then \( A \) is invertible and \( \text{det}(A^{-1}) = \frac{1}{5} \). (j) If \( A^2 = 0 \), then \( A \) is singular. ### Detailed Explanations: * **Statement (a):** - **Analysis:** For the products \( AB^\top \) and \( A^\top B \) to be defined, the inner dimensions of the resulting matrices must match. Both matrices \( A \) and \( B \) are \( m \times n \), so \( B^\top \) (the transpose of \( B \)) would be \( n \times m \). Thus, \( AB^\top \) would result in an \( m \times m \) matrix, and \( A^\top \) would be \( n \times m \) while \( B \) remains \( m \times n
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