Question

Find a 2x1 matrix x with entries not all zero such that Ax=3x, where A = [2 1; 1 2].

Am I supposed to use the Gaussian elimination method first? I tried it but I can't make matrix A into row echelon form.. can someone please help me?

Answer 1

Just let your matrix \[x=\left[\begin{matrix}x_1\\x_2\end{matrix}\right]\] and multiply your two matrices. Once you do that, solve for the values so that you get 3x.

Answer 2

Your solution should be something like \[Ax=\left[\begin{matrix} 2x_1+x_2 \\x_1+2x_2\end{matrix}\right]\]And you want \[2x_1+x_2=3x_1\]\[x_1+2x_2=3x_2\]With this you can do some quick substitutions and get an answer.

Answer 3

Thanks a lot! Is this the only way to solve this though? How I tried was subtract both sides of Ax=3x by 3x so you get Ax-3x=0 which becomes (A-3I)x=0. But then it didn't work..

Answer 4

I think your mistake in that was saying that 3x=3Ix. To have the identity matrix, you need it to be square, and x is not a square matrix.

Answer 5

what he has should still lead to a solution. Once you have\[(A-3I)x=0\]You should realize you are looking for a basis for the Null Space (or Kernel) of the matrix A-3I

Answer 6

Oh I see. So for similar questions like these, you would use the same techniques? I have another question that's like: Find a 3x1 matrix x with entries not all zero such that Ax=1x, where A = [1 2 -1; 1 0 1; 4 -4 5]. So I would set up x = [x1, x2, x3]?

Answer 7

he66666, do you know about eigenvalues or eigenvectors yet?

Answer 8

@joemath314159 We haven't learnd Null space yet so I don't really know..

Answer 9

No, we didn't learn about those either. We just started learning solving linear systems using the Gaussian elimination method.

Answer 10

wow. Your professor is very sneaky then lol. Since gaussian elimination is all you have, KingGeorge's method is best.

Answer 11

Once you learn things like null space, eigenvalues and whatnot, you should come back to this problem to have a good laugh.

Answer 12

Oh haha. I guess I'll use this technique to solve these problems. Thanks for your help joemath314159 and KingGeorge :)

Answer 13

You're welcome.