Question

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Short answer

The equation \(Ax = b\) has a solution for each b in \({\mathbb{R}^n}\).

Step-by-step solution

Show that the equation \[Ax = {\mathop{\rm b}\nolimits} \] has only a trivial solution

If \(Ax = 0\) has only a trivial solution, A must have a pivot in each of its \(n\) columns.

Explain that the equation Ax = b has exactly 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.

Since Ais a square matrix, itmust have a pivot in each row. According to theorem 4, the equation \(Ax = b\) has a solution for each b in \({\mathbb{R}^n}\).

Thus, the equation \(Ax = b\) has a solution for each b in \({\mathbb{R}^n}\).