Linear Algebra and Its Applications, Review Exercise 2.24

Review exercise 2.24. Suppose that A is a 3 by 5 matrix with the elementary vectors e_1, e_2, and e_3 in its column space. Does A has a left inverse? A right inverse?

Answer: Since e_1, e_2, and e_3 are in the column space the dimension of the column space must be 3 (since e_1, e_2, and e_3 are linearly independent) and thus the rank of A is r = 3 = m, the number of rows of A.

Since the rank of A equals the number of rows A has a 5 by 3 right inverse C. However it does not have a left inverse B since the rank r = 3 is less than the number of columns n = 5.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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