Linear Algebra Systems of Equations

[lina29]

Homework Statement

Consider the following two system of equations:

y_{1}=-2x_{1}-x_{2}+2x_{3}
y_{2}=2x_{1}+2x_{2}-3x_{3}
y_{3}=-2x_{1}-2x_{2}+2x_{3}

and

z_{1}=3y_{1}-4y_{2}-3y_{3}
z_{1}=3y_{1}-y_{2}-4y_{3}

Rewrite these 2 systems as Ax=y and By=z. Use this to get C so that Cx=z.
a) What is the matrix C?

B) Find the RREF matrix D which is row equivalent to the augmented matrix [C|z]

The Attempt at a Solution

My initial thought was that the matrix A would be:
-2 -1 2
2 2 -3
-2 -2 2

and matrix B:
3 -4 -3
3 -1 -4
and that C would be the product of A and B. However, I realized that since the column of A isn't the same as the row of B I couldn't form a product with them. I'm lost on how to find the matrix C. Do I need to invert a matrix in order to find C (the product)?

 

[HallsofIvy]

No, you don't. Your have Ax= y and By= z. Replace y in the second equation by Ax.

 

[lina29]

I'm sorry I don't understand what you mean. Are you saying the matrix A is right but I would have to substitute matrix B with Ax instead of y? I don't get how I would replace y with Ax. My thought is that i would multiply the coefficients of y in the matrix b with the coefficients of x in matrix a.

 

[BruceW]

I don't get how I would replace y with Ax.

Well, you had y=Ax and z=By, so replacing y with Ax gives: z=BAx. I think you came across problems because you tried to calculate AB (which is the wrong order).