Question

Determine whether the statements that follow are true or false, and justify your answer.

43.If two matrices A and B have the same reduced row-echelon form, then the equations $\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{B}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}$ must have the same solutions.

Short answer

The statement, “If two matrices A and B have the same reduced row-echelon form, then the equations $A\overrightarrow{x}=0andB\overrightarrow{x}=0$ must have the same solutions.” is true.

Step-by-step solution

Consider the condition 

The number of leading in the reduced row-echelon form of a matrix determines the number of solutions. The number of leading 1's is equal to the number of solutions of the linear system of equations.

Conclusion

So, the matrices A and B have the same reduced row-echelon form. Thus, the equations $A\overrightarrow{x}=0andB\overrightarrow{x}=0$ must have the same solutions.