Exercise 2.4.7. If 
is an 
by 
matrix with rank 
answer the following:
a) When is invertible, with 
existing such that 
?
b) When does 
have an infinite number of solutions for any 
?
Answer: a) Per theorem 2Q on page 96, for to have a right inverse 
such that 
we must have 
and 
. Per the same theorem, for to have a left inverse 
such that 
we must have 
and 
. For to have both a left and right inverse we must therefore have 
, in which case 
.
b) Per theorem 2Q on page 96, for to have at least one solution we must have . For to have an infinite number of solutions there must be at least one free variable, i.e., the rank must be less than the number of columns . We therefore must have and 
.
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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