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Q89E

Expert-verifiedFound in: Page 147

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**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.**

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

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

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

Now, since both *A* 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 $A=rref\left(M\right)$.

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