Linear Algebra and Its Applications, Exercise 2.4.17

Posted on August 6, 2012 by hecker

Exercise 2.4.17. Point out the error in the following argument: Suppose $B$ is a right-inverse of $A$ so that $AB = I$ . Then we can multiply both sides by $A^T$ to produce $A^TAB = A^T$ or $B = (A^TA)^{-1}A^T$ . But we then have $BA = (A^TA)^{-1}A^TA = I$ so that $B$ is a left-inverse as well.

Answer: The problem is that we have no guarantee that $(A^TA)^{-1}$ actually exists.

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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