Convert The Following Matrix A=\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 5 & 0 \\ 3 & 0 & 5\end{array}\right ] To Row Echelon Form.Options:A. B=\left(\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & 32\end{array}\right ]B.

Introduction

In linear algebra, a matrix is said to be in row echelon form (REF) if it satisfies certain conditions. The main goal of this article is to convert the given matrix $A$ to row echelon form. We will use the elementary row operations to achieve this.

Understanding Row Echelon Form

A matrix is in row echelon form if it satisfies the following conditions:

1. All the entries below the leading entry (or pivot) in each row are zeros.
2. The leading entry (or pivot) in each row is to the right of the leading entry in the row above it.
3. The leading entry (or pivot) in each row is a nonzero number.

The Given Matrix

The given matrix is:

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 2 & 5 & 0 \\ 3 & 0 & 5\end{array}\right)$$

Step 1: Apply Elementary Row Operations

To convert the matrix $A$ to row echelon form, we will apply the following elementary row operations:

• Swap two rows: Swap rows $i$ and $j$.
• Multiply a row by a scalar: Multiply row $i$ by a scalar $k$.
• Add a multiple of one row to another: Add $k$ times row $i$ to row $j$.

We will start by applying the first row operation to the matrix $A$.

Step 1.1: Apply the First Row Operation

We will apply the first row operation to the matrix $A$ by swapping rows 2 and 3.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 0 & 5 \\ 2 & 5 & 0\end{array}\right)$$

Step 1.2: Apply the Second Row Operation

We will apply the second row operation to the matrix $A$ by multiplying row 1 by 1 and row 2 by 1.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 0 & 5 \\ 2 & 5 & 0\end{array}\right)$$

Step 1.3: Apply the Third Row Operation

We will apply the third row operation to the matrix $A$ by adding -2 times row 1 to row 3.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 3 & 0 & 5 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.4: Apply the Fourth Row Operation

We will apply the fourth row operation to the matrix $A$ by multiplying row 2 by 1/3.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 1 & 0 & 5/3 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.5: Apply the Fifth Row Operation

We will apply the fifth row operation to the matrix $A$ by adding -1 times row 2 to row 1.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 1 & 0 & 5/3 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.6: Apply the Sixth Row Operation

We will apply the sixth row operation to the matrix $A$ by multiplying row 1 by 1 and row 3 by 1.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 1 & 0 & 5/3 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.7: Apply the Seventh Row Operation

We will apply the seventh row operation to the matrix $A$ by adding -1 times row 1 to row 2.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 0 & -2 & -4/3 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.8: Apply the Eighth Row Operation

We will apply the eighth row operation to the matrix $A$ by multiplying row 2 by -1/2.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 2/3 \\ 0 & 1 & -6\end{array}\right)$$

Step 1.9: Apply the Ninth Row Operation

We will apply the ninth row operation to the matrix $A$ by adding -1 times row 2 to row 3.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 2/3 \\ 0 & 0 & -34/3\end{array}\right)$$

Step 1.10: Apply the Tenth Row Operation

We will apply the tenth row operation to the matrix $A$ by multiplying row 3 by -3/34.

$$A=\left(\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 2/3 \\ 0 & 0 & 1\end{array}\right)$$

Step 1.11: Apply the Eleventh Row Operation

We will apply the eleventh row operation to the matrix $A$ by adding -2 times row 3 to row 1.

$$A=\left(\begin{array}{lll}1 & 2 & 1 \\ 0 & 1 & 2/3 \\ 0 & 0 & 1\end{array}\right)$$

Step 1.12: Apply the Twelfth Row Operation

We will apply the twelfth row operation to the matrix $A$ by adding -3/2 times row 3 to row 2.

$$A=\left(\begin{array}{lll}1 & 2 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$$

Step 1.13: Apply the Thirteenth Row Operation

We will apply the thirteenth row operation to the matrix $A$ by adding -2 times row 2 to row 1.

$$A=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$$

Step 1.14: Apply the Fourteenth Row Operation

We will apply the fourteenth row operation to the matrix $A$ by adding -1 times row 2 to row 1.

$$A=\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$$

Step 1.15: Apply the Fifteenth Row Operation

We will apply the fifteenth row operation to the matrix $A$ by adding -1 times row 3 to row 1.

$$A=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$$

Conclusion

We have successfully converted the matrix $A$ to row echelon form using the elementary row operations. The final matrix is:

$$B=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$$

This matrix is in row echelon form because it satisfies the conditions:

1. All the entries below the leading entry (or pivot) in each row are zeros.
2. The leading entry (or pivot) in each row is to the right of the leading entry in the row above it.
3. The leading entry (or pivot) in each row is a nonzero number.

Therefore, the correct answer is:

A. $B=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$

Final Answer

Introduction

In our previous article, we discussed how to convert a matrix to row echelon form using elementary row operations. In this article, we will answer some frequently asked questions related to converting a matrix to row echelon form.

Q: What is row echelon form?

A: Row echelon form is a form of a matrix where all the entries below the leading entry (or pivot) in each row are zeros. The leading entry (or pivot) in each row is to the right of the leading entry in the row above it. The leading entry (or pivot) in each row is a nonzero number.

Q: How do I convert a matrix to row echelon form?

A: To convert a matrix to row echelon form, you can use the following steps:

1. Swap two rows: Swap rows $i$ and $j$.
2. Multiply a row by a scalar: Multiply row $i$ by a scalar $k$.
3. Add a multiple of one row to another: Add $k$ times row $i$ to row $j$.

You can repeat these steps until the matrix is in row echelon form.

Q: What are the conditions for a matrix to be in row echelon form?

A: A matrix is in row echelon form if it satisfies the following conditions:

1. All the entries below the leading entry (or pivot) in each row are zeros.
2. The leading entry (or pivot) in each row is to the right of the leading entry in the row above it.
3. The leading entry (or pivot) in each row is a nonzero number.

Q: Can a matrix have multiple row echelon forms?

A: Yes, a matrix can have multiple row echelon forms. However, all the row echelon forms of a matrix are equivalent.

Q: How do I determine if a matrix is in row echelon form?

A: To determine if a matrix is in row echelon form, you can check the following conditions:

1. All the entries below the leading entry (or pivot) in each row are zeros.
2. The leading entry (or pivot) in each row is to the right of the leading entry in the row above it.
3. The leading entry (or pivot) in each row is a nonzero number.

If the matrix satisfies these conditions, then it is in row echelon form.

Q: Can a matrix be in both row echelon form and reduced row echelon form?

A: No, a matrix cannot be in both row echelon form and reduced row echelon form. A matrix can be in either row echelon form or reduced row echelon form, but not both.

Q: What is the difference between row echelon form and reduced row echelon form?

A: The main difference between row echelon form and reduced row echelon form is that in reduced row echelon form, all the entries above the leading entry (or pivot) in each row are zeros.

Q: How do I convert a matrix from row echelon form to reduced row echelon form?

A: To convert a matrix from row echelon form to reduced row echelon form, you can use the following steps:

1. Add a multiple of one row to another: Add $k$ times row $i$ to row $j$.
2. Multiply a row by a scalar: Multiply row $i$ by a scalar $k$.

You can repeat these steps until the matrix is in reduced row echelon form.

Conclusion

In this article, we answered some frequently asked questions related to converting a matrix to row echelon form. We discussed the conditions for a matrix to be in row echelon form, how to convert a matrix to row echelon form, and the difference between row echelon form and reduced row echelon form. We also provided some tips on how to determine if a matrix is in row echelon form and how to convert a matrix from row echelon form to reduced row echelon form.

Final Answer

The final answer is that converting a matrix to row echelon form is an important concept in linear algebra, and understanding the conditions for a matrix to be in row echelon form and how to convert a matrix to row echelon form is crucial for solving systems of linear equations.