Question

Let A be a 4 × 4 matrix, and let$\overrightarrow{\mathbf{b}}$and$\overrightarrow{\mathbf{c}}$ be two vectors in${\mathbf{ℝ}}^{\mathbf{4}}$ . We are told that the system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}$ has a unique solution. What can you say about the number of solutions of the system$\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{c}}$ ?

Short answer

The system $\mathrm{A}\overrightarrow{\mathrm{x}}=\overrightarrow{\mathrm{c}}$will have a unique solution, if the system $\mathrm{A}\overrightarrow{\mathrm{x}}=\overrightarrow{b}$has unique solution.

Step-by-step solution

Consider the system.

A matrix in reduced row-echelon form is used to solve systems of linear equations. There are four prerequisites for the reduced row echelon form:

• The number 1 is the first non-zero integer in the first row (the leading entry).
• The second row begins with the number 1, which is more to the right than the first row's leading item. The number 1 must be further to the right in each consecutive row.
• Each row's first item must be the sole non-zero number in its column.
• Any rows that are not zero are pushed to the bottom of the matrix.

A is a 4 × 4 matrix, and $\overrightarrow{b}$,$\overrightarrow{c}$ are two vectors in ${\mathrm{ℝ}}^4$and the system $A\overrightarrow{x}=\overrightarrow{b}$has a unique solution.

Consider the reduced row-echelon form

As the system $A\overrightarrow{x}=\overrightarrow{b}$has unique solution, thus, the reduced row-echelon form of the matrix A will be,

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

As $\overrightarrow{b}$and $\overrightarrow{c}$are the vectors in ${\mathrm{ℝ}}^4$, thus, if $A\overrightarrow{x}=\overrightarrow{b}$has a unique solution, then $A\overrightarrow{x}=\overrightarrow{c}$will have unique solution.

Final answer.

As the system $A\overrightarrow{x}=\overrightarrow{b}$has unique solution, thus, the system $A\overrightarrow{x}=\overrightarrow{c}$will have unique solution.