Number 15 through 28 ### Linear Algebra: Advanced Problem Set**15.** Is it possible for a $ 4 \times 4 $ matrix to be invertible when its columns do not span $ \mathbb{R}^4 $? Why or why not?**16.** If an $ n \times n $ matrix $ A $ is invertible, then the columns of $ A^T $ are linearly independent. Explain why.**17.** Can a square matrix with two identical columns be invertible? Why or why not?**18.** Can a square matrix with two identical rows be invertible? Why or why not?**19.** If the columns of a $ 7 \times 7 $ matrix $ D $ are linearly independent, what can be said about the solutions of $ Dx = b $? Why?**20.** If $ A $ is a $ 5 \times 5 $ matrix and the equation $ Ax = b $ is consistent for every $ b $ in $ \mathbb{R}^5 $, is it possible that for some $ b $, the equation $ Ax = b $ has more than one solution? Why or why not?**21.** If the equation $ C u = v $ has more than one solution for some $ v $ in $ \mathbb{R}^n $, can the columns of the $ n \times n $ matrix $ C $ span $ \mathbb{R}^n $? Why or why not?**22.** If $ n \times n $ matrices $ E $ and $ F $ have the property that $ EF = I $, then $ E $ and $ F $ commute. Explain why.**23.** Assume that $ F $ is an $ n \times n $ matrix. If the equation $ F x = y $ is inconsistent for some $ y $ in $ \mathbb{R}^n $, what can you say about the equation $ F x = 0 $? Why?**24.** If an $ n \times n $ matrix $ G $ cannot be row reduced to $ I_n $, what can you say about the columns of $ G $? Why?**25.** Verify the boxed statement preceding Example 1.**

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Description: Number 15 through 28 ### Linear Algebra: Advanced Problem Set**15.** Is it possible for a $ 4 \times 4 $ matrix to be invertible when its columns do not span $ \mathbb{R}^4 $? Why or why not?**16.** If an $ n \times n $ matrix $ A $ is invert

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15. Is it possible for a 4 × 4 matrix to be invertible when its columns do not span R*? Why or why not? 16. If an n x n matrix A is invertible, then the columns of AT are linearly independent. Explain why. 17. Can a square matrix with two identical columns be invert- ible? Why or why not?

15. Is it possible for a 4 × 4 matrix to be invertible when its columns do not span R*? Why or why not? 16. If an n x n matrix A is invertible, then the columns of AT are linearly independent. Explain why. 17. Can a square matrix with two identical columns be invert- ible? Why or why not?

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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Question

Number 15 through 28

### Linear Algebra: Advanced Problem Set

**15.** Is it possible for a $ 4 \times 4 $ matrix to be invertible when its columns do not span $ \mathbb{R}^4 $? Why or why not?

**16.** If an $ n \times n $ matrix $ A $ is invertible, then the columns of $ A^T $ are linearly independent. Explain why.

**17.** Can a square matrix with two identical columns be invertible? Why or why not?

**18.** Can a square matrix with two identical rows be invertible? Why or why not?

**19.** If the columns of a $ 7 \times 7 $ matrix $ D $ are linearly independent, what can be said about the solutions of $ Dx = b $? Why?

**20.** If $ A $ is a $ 5 \times 5 $ matrix and the equation $ Ax = b $ is consistent for every $ b $ in $ \mathbb{R}^5 $, is it possible that for some $ b $, the equation $ Ax = b $ has more than one solution? Why or why not?

**21.** If the equation $ C u = v $ has more than one solution for some $ v $ in $ \mathbb{R}^n $, can the columns of the $ n \times n $ matrix $ C $ span $ \mathbb{R}^n $? Why or why not?

**22.** If $ n \times n $ matrices $ E $ and $ F $ have the property that $ EF = I $, then $ E $ and $ F $ commute. Explain why.

**23.** Assume that $ F $ is an $ n \times n $ matrix. If the equation $ F x = y $ is inconsistent for some $ y $ in $ \mathbb{R}^n $, what can you say about the equation $ F x = 0 $? Why?

**24.** If an $ n \times n $ matrix $ G $ cannot be row reduced to $ I_n $, what can you say about the columns of $ G $? Why?

**25.** Verify the boxed statement preceding Example 1.

**

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