## 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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