## 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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