Question

Suppose Ais 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.

Short answer

The equation Ax = b has a unique solution.

Step-by-step solution

State that the equation Ax = b has at least one solution

Theorem 4states 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.

Explain that equation Ax = b has exactly one solution

Since Ais a square matrix, ithas 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.