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Chapter 2: Q2.3-29Q (page 93)

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.

Short Answer

Expert verified

The transformation is not invertible.

Step by step solution

01

State the invertible matrix theorem

The statements are identical according to the invertible matrix theorem, as shown below:

  1. The matrix is invertible.
  2. For some b, the matrix equation\[A{\bf{x}} = {\bf{b}}\]does not have a unique solution (more than one solution).
  3. The linear transformation\(x| \to Ax\)is one-to-one.
  4. The mapping of \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\) is equivalent to the linear transformation \(x| \to Ax\).
02

Define the transformation

According to the invertible matrix theorem, if the matrix equation\[A{\bf{x}} = {\bf{b}}\]has more than one solution, the transformation is one-to-one.

From the given statement, the matrix equation has more than one solution for some b, but the linear transformation\(x| \to Ax\)is not one-to-one.

So, the given statement cannot be true.

Therefore, both the matrix and the transformation are not invertible.

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