Question

If \(C\) is \({\bf{6}} \times {\bf{6}}\) and the equation \(C{\bf{x}} = {\bf{v}}\) is consistent for every \({\bf{v}}\) in \({\mathbb{R}^{\bf{6}}}\), is it possible that for some \({\bf{v}}\), the equation \(C{\bf{x}} = {\bf{v}}\) has more than one solution? Why or why not?

Short answer

The given equation has a unique solution.

Step-by-step solution

Find the existence of the inverse of the matrix

Matrix \(C\) of the order \(6 \times 6\) is invertibleas \(C{\bf{x}} = {\bf{v}}\) is consistent for each value of v in\({\mathbb{R}^6}\).

Find the nature of the solution of the equation \(C{\bf{x}} = {\bf{b}}\)

Since \(C\) is invertible, for each value of \({\bf{v}}\) in \({\mathbb{R}^6}\), the equation \(C{\bf{x}} = {\bf{v}}\) has a unique solution as given by the equation.

\({\bf{x}} = {C^{ - 1}}{\bf{v}}\)

So, the equation \(C{\bf{x}} = {\bf{v}}\) has a unique solution.