True or False? Let A be an n x n matrix and S is an n x n invertible matrix. If x is a solution to the system (S⁻¹AS)x = b, then Sx is a solution to the system Ay = Sb.
Jun 5, 5:30pm

Let's analyze the given statement step-by-step.

We are given:

  • AAA is an n×nn \times nn×n matrix.
  • SSS is an invertible n×nn \times nn×n matrix.
  • xxx is a solution to the system (S−1AS)x=b(S^{-1}AS)x = b(S−1AS)x=b.

We need to determine if SxSxSx is a solution to the system Ay=SbAy = SbAy=Sb.

  1. Start with the given equation:
(S−1AS)x=b (S^{-1}AS)x = b(S−1AS)x=b
  1. Multiply both sides by SSS to eliminate S−1S^{-1}S−1:
S(S−1AS)x=Sb S(S^{-1}AS)x = SbS(S−1AS)x=Sb

Simplifying the left side:

(SS−1)ASx=ASx (S S^{-1})ASx = ASx(SS−1)ASx=ASx

Since SS−1=IS S^{-1} = ISS−1=I (the identity matrix), we have:

ASx=Sb ASx = SbASx=Sb
  1. Now, let y=Sxy = Sxy=Sx. Substitute yyy into the equation:
Ay=Sb Ay = SbAy=Sb

Thus, y=Sxy = Sxy=Sx satisfies the equation Ay=SbAy = SbAy=Sb. Therefore, the statement is true.

Jun 5, 5:30pm

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