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Expert-verified Found in: Page 1 ### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384 # If A is 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.a

The transformation is not invertible.

See the step by step solution

## Step 1: 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 $$x| \to Ax$$.

## Step 2: 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. ### Want to see more solutions like these? 