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ང་ ? ? ? ? ? Are the following statements true or false? Here A is an n x n square matrix. ? 1. If A and B are both invertible then A + B is also invertible. ? 2. An invertible matrix A can be written as a product of elementary matrices in a unique way. ? ◇ 3. The zero matrix is invertible. ? 4. If A is not invertible then the system Ar : = b has infinitely many solutions. ? ☐ 5. If A is invertible then A # 0. ? ✰ 6. If the span of the rows of A is R" then A is invertible. <> 7. If A is the inverse of B then A and B commute. ✰ 8. The product of two elementary matrices is an elementary matrix. ? 9. If A0 then A is invertible. ? ◇ 10. If A² is invertible then A is invertible. 11. A diagonal matrix is invertible if all of the entries on the diagonal are non-zero. ◇ 12. If A is invertible then so is AT. 13. If A = 0 has only the trivial solution then Ar= = b has a unique solution for all choices of b in R".

ང་ ? ? ? ? ? Are the following statements true or false? Here A is an n x n square matrix. ? 1. If A and B are both invertible then A + B is also invertible. ? 2. An invertible matrix A can be written as a product of elementary matrices in a unique way. ? ◇ 3. The zero matrix is invertible. ? 4. If A is not invertible then the system Ar : = b has infinitely many solutions. ? ☐ 5. If A is invertible then A # 0. ? ✰ 6. If the span of the rows of A is R" then A is invertible. <> 7. If A is the inverse of B then A and B commute. ✰ 8. The product of two elementary matrices is an elementary matrix. ? 9. If A0 then A is invertible. ? ◇ 10. If A² is invertible then A is invertible. 11. A diagonal matrix is invertible if all of the entries on the diagonal are non-zero. ◇ 12. If A is invertible then so is AT. 13. If A = 0 has only the trivial solution then Ar= = b has a unique solution for all choices of b in R".

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 24E
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ང་ ? ? ? ? ? Are the following statements true or false? Here A is an n x n square matrix. ? 1. If A and B are both invertible then A + B is also invertible. ? 2. An invertible matrix A can be written as a product of elementary matrices in a unique way. ? ◇ 3. The zero matrix is invertible. ? 4. If A is not invertible then the system Ar : = b has infinitely many solutions. ? ☐ 5. If A is invertible then A # 0. ? ✰ 6. If the span of the rows of A is R" then A is invertible. <> 7. If A is the inverse of B then A and B commute. ✰ 8. The product of two elementary matrices is an elementary matrix. ? 9. If A0 then A is invertible. ? ◇ 10. If A² is invertible then A is invertible. 11. A diagonal matrix is invertible if all of the entries on the diagonal are non-zero. ◇ 12. If A is invertible then so is AT. 13. If A = 0 has only the trivial solution then Ar= = b has a unique solution for all choices of b in R".
Transcribed Image Text:ང་ ? ? ? ? ? Are the following statements true or false? Here A is an n x n square matrix. ? 1. If A and B are both invertible then A + B is also invertible. ? 2. An invertible matrix A can be written as a product of elementary matrices in a unique way. ? ◇ 3. The zero matrix is invertible. ? 4. If A is not invertible then the system Ar : = b has infinitely many solutions. ? ☐ 5. If A is invertible then A # 0. ? ✰ 6. If the span of the rows of A is R" then A is invertible. <> 7. If A is the inverse of B then A and B commute. ✰ 8. The product of two elementary matrices is an elementary matrix. ? 9. If A0 then A is invertible. ? ◇ 10. If A² is invertible then A is invertible. 11. A diagonal matrix is invertible if all of the entries on the diagonal are non-zero. ◇ 12. If A is invertible then so is AT. 13. If A = 0 has only the trivial solution then Ar= = b has a unique solution for all choices of b in R".
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