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### 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

# Suppose A is an $$n \times n$$ matrix with the property that the equation $$A{\mathop{\rm x}\nolimits} = 0$$ has at least one solution for each b in $${\mathbb{R}^n}$$. Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.

The equation Ax = b has a unique solution.

See the step by step solution

## Step 1: State that the equation Ax = b has at least one solution

Theorem 4 states that if $$A$$ is an $${\rm{m}} \times n$$ matrix, then the following statements are equivalent.

1. For each $${\mathop{\rm b}\nolimits}$$ in $${\mathbb{R}^m}$$, the equation $$Ax = b$$ has a solution.
2. Each $${\mathop{\rm b}\nolimits}$$ in $${\mathbb{R}^m}$$ is a linear combination of the columns of A.
3. The columns of $$A$$ span .
4. $$A$$ has a pivot position in every row.

By theorem 4, matrix A has a pivot position in each row because the equation Ax = b has a solution for each b.

## Step 2: Explain that equation Ax = b has exactly one solution

Since A is a square matrix, it has a pivot in each column. The equation Ax = b has no free variables, which demonstrates that the solution is unique.

Thus, the equation Ax = b has a unique solution.