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Q89E

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Found in: Page 147

### Linear Algebra With Applications

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${\mathbf{}}{\mathbit{A}}{\mathbf{=}}{\mathbit{r}}{\mathbit{r}}{\mathbit{e}}{\mathbit{f}}{\mathbf{\left(}}{\mathbit{M}}{\mathbf{\right)}}$. Hint: Both A and ${\mathbf{}}{\mathbit{r}}{\mathbit{r}}{\mathbit{e}}{\mathbit{f}}{\mathbf{\left(}}{\mathbit{M}}{\mathbf{\right)}}$are in reduced row-echelon form, and they have the same kernel. Exercise 88 is helpful.

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

See the step by step solution

## Step 1: Given in the question.

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

## Step 2: 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 both A and $rref\left(M\right)$ are reduced row-echelon form.

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

Thus, the matrices are same.

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