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Determine if each statement below is true or false. 1. Suppose AB AC. Then B = C. 2. (A+ B)" = A" + B" %3D 3. A(BC) = (AC)B

Determine if each statement below is true or false. 1. Suppose AB AC. Then B = C. 2. (A+ B)" = A" + B" %3D 3. A(BC) = (AC)B

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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Determine if each statement below is true or false. 1. Suppose AB = AC. Then B = C. 2. (A+ B)" = A" + B" %3D 3. A(BC) = (AC)B 4. (AB)- = A-!B-1 5. Suppose AB = AC and A is invertible. Then B = C. 6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R". 7. If the columns of an n x n matrix A span R", then the columns of A are linearly independent. 8. If the equation Ax = b has more than one solution for some b in R", then the columns of A span R". 9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. 10. Every square triangular matrix is invertible. 11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. 14. The dimension of Col(A) is the number of pivot columns of A. 15. Col(A) is the set of solutions to Ax = b.
Transcribed Image Text:Determine if each statement below is true or false. 1. Suppose AB = AC. Then B = C. 2. (A+ B)" = A" + B" %3D 3. A(BC) = (AC)B 4. (AB)- = A-!B-1 5. Suppose AB = AC and A is invertible. Then B = C. 6. If A is an invertible n x n matrix, then Ax = b is consistent for all b in R". 7. If the columns of an n x n matrix A span R", then the columns of A are linearly independent. 8. If the equation Ax = b has more than one solution for some b in R", then the columns of A span R". 9. If A is an n x n matrix and the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivots. 10. Every square triangular matrix is invertible. 11. Every line in R" is a subspace of R". 12. Every line through the origin in R" is a subspace of R". 13. The dimension of Nul(A) is the number of variables in the equation Ax = 0. 14. The dimension of Col(A) is the number of pivot columns of A. 15. Col(A) is the set of solutions to Ax = b.
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