Hope has rewritten a system of equations as the following matrix:
	2	–1	–3	
1	2	–4

Now, Hope wants to convert this matrix into row echelon form.

Which of the following can she do first?

Select THREE correct answers.
A.She can switch rows #1 and #2.
B.She can multiply row #2 by −2.
C.She can add −1 to each element of row #2.
D.She can subtract each element of row #2 from its corresponding element in row #1.
E.She can switch the last element of row #1 with its corresponding element in row #2.

Hope has the following matrix:

\[
\begin{pmatrix}
2 & -1 & -3 \\
1 & 2 & -4
\end{pmatrix}
\]

To convert this matrix into **row echelon form**, the goal is to get leading 1's and zeros below the leading entries (pivots). The following steps are possible in this process:

### Let’s examine the options one by one:

1. **Option A: She can switch rows #1 and #2.**
   - **Correct**. Row switching is a valid elementary row operation in Gaussian elimination. It can be helpful to get a leading 1 in the first row. By switching rows, Hope will place a 1 (from the second row) at the top.
   
2. **Option B: She can multiply row #2 by −2.**
   - **Correct**. Multiplying a row by a non-zero scalar is a valid elementary row operation. This can help create a more convenient form, such as getting rid of fractions or simplifying a row.

3. **Option C: She can add −1 to each element of row #2.**
   - **Incorrect**. Adding the same constant to every element in a row is **not** a valid elementary row operation. Instead, Hope can add or subtract rows or multiples of rows from each other.

4. **Option D: She can subtract each element of row #2 from its corresponding element in row #1.**
   - **Correct**. This operation is valid because subtracting one row from another is a standard row operation. It can help eliminate terms to create zeros.

5. **Option E: She can switch the last element of row #1 with its corresponding element in row #2.**
   - **Incorrect**. Swapping individual elements within a matrix is **not** a valid elementary row operation in Gaussian elimination.

### The correct answers are:
- **A.** She can switch rows #1 and #2.
- **B.** She can multiply row #2 by −2.
- **D.** She can subtract each element of row #2 from its corresponding element in row #1.

Would you like further details on how row echelon form is achieved or any other aspect? 

---

Here are 5 related questions for further practice:

1. How can Hope use row operations to simplify a 3x3 matrix?
2. What are the steps for solving a system of equations using Gaussian elimination?
3. Can Hope use a row operation to create a leading 1 in any row?
4. What is the difference between row echelon form and reduced row echelon form?
5. How does Hope identify when her matrix is in row echelon form?

**Tip:** Always aim to get a leading 1 in each row, starting from the top-left, and create zeros below it as you work through the matrix.

Hope has rewritten a system of equations as the following matrix:
2 -1 -3
1 2 -4

Now, Hope wants to convert this matrix into row echelon form.
Which of the following can she do first?

Select THREE correct answers.
A.She can switch rows #1 and #2.
B.She can multiply row #2 by −2.
C.She can add −1 to each element of row #2.
D.She can subtract each element of row #2 from its corresponding element in row #1.
E.She can switch the last element of row #1 with its corresponding element in row #2.

Learn how to convert a 2x3 matrix into row echelon form using elementary row operations such as row switching, scalar multiplication, and row subtraction. This example involves a system of equations and is ideal for students in Grades 10-12 studying linear algebra and Gaussian elimination.

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