Exercise 1.6.20. Suppose is a 3 by 3 matrix for which the third row is the sum of the first and second rows. Show that there is no solution to the equation , and determine whether has an inverse or not.
Answer: For we have the following system of three equations:
If we subtract the sum of the first two equations from the third equation, we obtain the following:
Since we have reached a contradiction we conclude that the system of equations has no solutions. The corresponding matrix is singular and therefore has no inverse.
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.