Exercise 1.3.9. State whether the following statements are true or false. (Note that without loss of generality we can assume that no row exchanges occur during the process of elimination.)

(a) Given a system in u, v, etc., where the third equation starts with zero (i.e., u has coefficient 0), during the process of elimination no multiple of the first equation will be subtracted from the third equation.

(b) Given a system in u, v, etc., where the third equation has zero as the second coefficient (i.e., v has coefficient 0), during the process of elimination no multiple of the second equation will be subtracted from the third equation.

(c) Given a system in u, v, etc., where the third equation has zero as both the first and the second coefficient (i.e., both u and v have coefficient 0), during the process of elimination no multiple of either the first or the second equation will be subtracted from the third equation.

Answer: (a) True. Since the coefficient of u in the third equation is already zero, in the first step of elimination there is no need to subtract a multiple of the first equation in order to transform the third equation into a new equation with a zero coefficient for u.

(b) False. Since the first coefficient (of u) of the third equation may be nonzero, in the first step of elimination we may need to subtract a multiple of the first equation from the third equation in order to transform the third equation into a new equation with a zero coefficient for u. In that case, if the second coefficient in the first equation is nonzero then that will result in the second coefficient (of v) in the third equation becoming nonzero.

In that event, in the next step of elimination we would need to subtract a multiple of the second equation from the third equation in order to transform the third equation into a new equation with a zero coefficient for v.

(c) True. As in (a), since the coefficient of u in the third equation is already zero, in the first step of elimination there is no need to subtract a multiple of the first equation in order to transform the third equation into a new equation with a zero coefficient for u. In that case the coefficient of v in the third equation would remain zero, and there would be no need to subtract a multiple of the second equation in order to transform the third equation into a new equation with a zero coefficient for v.

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.