SOLUTION: Hello, I need help with the following problem: Three basis are given in the plane. With respect to those basis a point has as components (x1,x2), (y1,y2) and (z1,z2). Suppose th

Question 1206684: Hello, I need help with the following problem:
Three basis are given in the plane. With respect to those basis a point has as components (x1,x2), (y1,y2) and (z1,z2). Suppose that [y1 y2] = [x1 x2]A, [z1 z2] = [x1 x2]B and [z1 z2] = [y1 y2]C being A,B,C 2x2 matrices. Express C as a function of A and B.
I very much appreciate your help.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
We're given the following relationships:
1. [y1 y2] = [x1 x2]A
2. [z1 z2] = [x1 x2]B
3. [z1 z2] = [y1 y2]C
We want to express C as a function of A and B.
**Step 1: Substitute [y1 y2] in equation 3:**
From equation 1, we know that [y1 y2] = [x1 x2]A. Substitute this into equation 3:
[z1 z2] = ([x1 x2]A)C
**Step 2: Substitute [z1 z2] from equation 2:**
From equation 2, we know that [z1 z2] = [x1 x2]B. Substitute this into the previous equation:
[x1 x2]B = ([x1 x2]A)C
**Step 3: Simplify and Solve for C:**
Since [x1 x2] is a common factor on both sides, we can cancel it out:
B = AC
**Now, to isolate C, we can multiply both sides by the inverse of A (assuming A is invertible):**
A⁻¹B = C
**Therefore, C = A⁻¹B.**

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