Question

Suppose a matrix A in reduced row-echelon form can be obtained from a matrix M by a sequence of elementary row operations. Show that$\mathbf{}\mathit{A}\mathbf{=}\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{\left(}\mathit{M}\mathbf{\right)}$. Hint: Both A and $\mathbf{}\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{\left(}\mathit{M}\mathbf{\right)}$are in reduced row-echelon form, and they have the same kernel. Exercise 88 is helpful.

Short answer

Thus, it is proved that $A=rref\left(M\right)$.

Step-by-step solution

Given in the question.

BothA and$rref\left(M\right)$ are in reduced row-echelon form, and they have the same kernel.

Prove that A=rref(M).

Suppose A is in reduced row –echelon form which is obtained from a matrix M by a sequence of elementary row operations.

Now, since bothA and$rref\left(M\right)$ are reduced row-echelon form.

Therefore, they are the same matrices such that $ker\hspace{0.33em}A=ker\left(rref\right(M\left)\right)$.

Thus, the matrices are same.

Hence, it is proved that $A=rref\left(M\right)$.