Exercise 2.2.12. What is a 2 by 3 system of equations that has the following general solution?
Answer: The general solution above is the sum of a particular solution and a homogeneous solution, where
Since is the only variable referenced in the homogeneous solution it must be the only free variable, with and being basic. Since is basic we must have a pivot in column 1, and since is basic we must have a second pivot in column 2. After performing elimination on the resulting echelon matrix must therefore have the form
To simplify solving the problem we can assume that also has this form; in other words, we assume that is already in echelon form and thus we don’t need to carry out elimination. The matrix then has the form
where and are nonzero (because they are pivots).
We then have
If we assume that is 1 and express the right-hand side in matrix form this then becomes
or (expressed as a system of equations)
The pivot must be nonzero, and we arbitrarily assume that . We can then satisfy the first equation by assigning and . The pivot must also be nonzero, and we arbitrarily assume that as well. We can then satisfy the second equation by assigning . Our proposed value of is then
so that we have
We next turn to the general system . We now have a value for , and we were given the value of the particular solution. We can multiply the two to calculate the value of :
This gives us the following as an example 2 by 3 system that has the general solution specified above:
Finally, note that the solution provided for exercise 2.2.12 at the end of the book is incorrect. The right-hand side must be a 2 by 1 matrix and not a 3 by 1 matrix, so the final value of 0 in the right-hand side should not be present.
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.