Exercise 2.2.11. Let be a system of equations in unknowns, and suppose that the only solution to this system is . In this case what is the rank of ?
Answer: If the system has no solutions then the system has no free variables. If the system has no free variables then all variables must be basic variables. But the number of basic variables is the same as the number of pivots, and if all variables are basic then there must be a pivot in every column. The number of pivots is then equal to the number of columns and we have the rank .
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.