Question

Question:IfAandBare two2x2matrices such that the equations $\mathit{A}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{=}\mathbf{0}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{B}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{0}$ have the same solutions, then rref (A) must be equal to rref(B) .

Short answer

The statement, “If A and B are two 2x2 matrices such that the equations $A\stackrel{\rightharpoonup }{x}=0andB\overrightarrow{x}=0$ have the same solutions, then rref(A) must be equal to rref(B) .” is true.

Step-by-step solution

Consider the condition

When a matrix is said to be in its reduced row-echelon form, then, it represents the number of solutions present in it.

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

As the matrices A and B have same solutions, thus, rref(A) must be equal to rref(B) .”

Final answer

The matrices A and B of order 2x2 with equations $A\stackrel{\rightharpoonup }{x}=0andB\overrightarrow{x}=0$ having same solutions must have, .