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Problem 181

# Show that the matrix $$\mathrm{A}=\begin{array}{cll} 0 & 1 & 2 \\ 2 & 3 & 4 \\ 14 & 7 & 10 \end{array}$$ is equivalent to $$\mathrm{D}^{33}{ }_{2}$$ where $\mathrm{D}^{\mathrm{m}, \mathrm{n}} \mathrm{r}$ denotes the canonical form under equivalence of A. D $$^{\mathrm{m}, \mathrm{n}}_{\mathrm{r}}$$ is the echelon form that has one's along the diagonal and zeros elsewhere, and where all the zero rows are consigned to the depths of the matrix.

Expert verified
The matrix A, after performing elementary row operations, becomes: $\begin{array}{cll} 1 & 3/2 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & -9 \end{array}$ This result is not in the form D^(m,n)_r, and therefore, the given matrix A is not equivalent to D^(33)_2.
See the step by step solution

## Step 1: Identify the elementary row operations required

We will perform the following elementary row operations to bring the matrix A to its canonical form: 1. Swap Row 1 and Row 2. 2. Add (-7) times Row 1 to Row 3. 3. Multiply Row 1 by (1/2). 4. Add (-2) times Row 1 to Row 2. 5. Add Row 2 to Row 3.

## Step 2: Perform the elementary row operations

Let's perform the elementary row operations one by one on the matrix A. 1. Swap Row 1 and Row 2: $\begin{array}{cll} 2 & 3 & 4 \\ 0 & 1 & 2 \\ 14 & 7 & 10 \end{array}$ 2. Add (-7) times Row 1 to Row 3: $\begin{array}{cll} 2 & 3 & 4 \\ 0 & 1 & 2 \\ 0 & -14 & -18 \end{array}$ 3. Multiply Row 1 by (1/2): $\begin{array}{cll} 1 & 3/2 & 2 \\ 0 & 1 & 2 \\ 0 & -14 & -18 \end{array}$ 4. Add (-2) times Row 1 to Row 2: $\begin{array}{cll} 1 & 3/2 & 2 \\ 0 & 1 & 2 \\ 0 & -7 & -11 \end{array}$ 5. Add Row 2 to Row 3: $\begin{array}{cll} 1 & 3/2 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & -9 \end{array}$

## Step 3: Compare canonical form obtained with D^(m,n)_r

We can see that the canonical form of matrix A obtained after performing the elementary row operations is: $\begin{array}{cll} 1 & 3/2 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & -9 \end{array}$ Since this matrix is not in the form D^(m,n)_r, it cannot be said that the given matrix A is equivalent to D^(33)_2.

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