Question

I need to find a 3x3 matrix so that Ax=(1,2,1) has no solutions and both Ax=(1,1,0) and Ax=(1,0,1) have infinitly many solutions. Cannot seem to find anything in Strang's book that lets me know how to approach this, or any similar examples!

Answer 1

In section 3.4 of the book, under the title "The Complete Solution", you can find a listing of the four possibilities for linear equations (that depend on the rank r). For your particular problem, you should be looking at case-4 (r < m & r < n).

From the facts: Ax = (1 0 1) & Ax = (1 1 0) has infinitely many solutions, it is seen that b1 = b2 + b3. From this it is clear that (1 2 1) does not have any solution. A matrix such as the one below should solve the problem:
\[\left[\begin{matrix}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & -1 & 1\end{matrix}\right]\]

Answer 2

Another way to look at it.

Since (101) and (1 1 0) have infinitely many solutions, the matrix we're after is singular,
and since there are solutions at all, both (101) and (110) must be in the column space of A, so we can choose(for some a,b,c)

A= ( 1 1 a
1 0 b
0 1 c )

Since the matrix is singular, elimination must give us 0's on the bottom row, so

1 1 a 1 1 1 a 1 1 1 a 1
1 0 b 2 -> 0 -1 b-a 1 -> 0 -1 b-a 1
0 1 c 1 0 1 c 1 0 0 b+c-a 2

So, if b+c-a=0, we have 0=2 on the bottom row, and there is no solution. Choose
any b & c you want.......