## Linear Algebra and Its Applications, Exercise 2.4.7

Exercise 2.4.7. If $A$ is an $m$ by $n$ matrix with rank $r$ answer the following:

a) When is $A$ invertible, with $A^{-1}$ existing such that $AA^{-1} = A^{-1}A = I$?

b) When does $Ax = b$ have an infinite number of solutions for any $b$?

Answer: a) Per theorem 2Q on page 96, for $A$ to have a right inverse $C$ such that $AC = I$ we must have $r = m$ and $m \le n$. Per the same theorem, for $A$ to have a left inverse $B$ such that $BA = I$ we must have $r = n$ and $m \ge n$. For $A$ to have both a left and right inverse we must therefore have $r = m = n$, in which case $A^{-1} = B = C$.

b) Per theorem 2Q on page 96, for $Ax = b$ to have at least one solution we must have $r = m$. For $Ax = b$ to have an infinite number of solutions there must be at least one free variable, i.e., the rank $r$ must be less than the number of columns $n$.  We therefore must have $r = m$ and $m < n$.

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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