Question

Suppose an \(m \times n\) matrix Ahas \(n\) pivot columns. Explain why for each b in \({\mathbb{R}^m}\) the equation \(A{\bf{x}} = {\bf{b}}\) has at most one solution. [Hint:Explain why \(A{\bf{x}} = {\bf{b}}\) cannot have infinitely many solutions.]

Short answer

The equation \(A{\bf{x}} = {\bf{b}}\) has at most one solution.

Step-by-step solution

The condition for infinitely many solutions

For the matrix equation \(A{\bf{x}} = {\bf{b}}\), the system of equations has infinitely many solutions if the variables are free.

Explanation for why the matrix equation cannot have infinitely many solutions

For a matrix of order\(m \times n\), there are\(n\)columns in matrix A. Also, matrix A has\(n\)pivot columns, which means each column has a pivot position.

It shows that the matrix equation\(A{\bf{x}} = {\bf{b}}\)does not have free variables. Thus, the system of equations cannot have infinitely many solutions.

Hence, the equation has at most one solution.