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Q31Q

Expert-verifiedFound in: Page 1

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 *A*x = b has a unique solution.

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

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

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

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

Thus, the equation *A*x = b has a unique solution.

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