Use elementary transformation to reduce the following matrix A to triangular form and hence find the rank of A, Where 𝐴 = [ 1 2 3 0 2 4 3 2 3 2 1 3 6 8 7 5 ].

Let's reduce the matrix \( A \) to its triangular form using elementary row operations (also known as Gaussian elimination). Once it's in triangular form, we can determine the rank of the matrix by counting the number of non-zero rows.

### Given Matrix
The matrix \( A \) is:
\[
A = \begin{pmatrix}
1 & 2 & 3 & 0 \\
2 & 4 & 3 & 2 \\
3 & 2 & 1 & 3 \\
6 & 8 & 7 & 5
\end{pmatrix}
\]

### Step 1: Perform Elementary Row Operations to Achieve Triangular Form
1. **Make the first column's elements below the first row zero**:
   - Subtract 2 times Row 1 from Row 2 (R2 = R2 - 2R1):
     \[
     \begin{pmatrix}
     1 & 2 & 3 & 0 \\
     0 & 0 & -3 & 2 \\
     3 & 2 & 1 & 3 \\
     6 & 8 & 7 & 5
     \end{pmatrix}
     \]
   - Subtract 3 times Row 1 from Row 3 (R3 = R3 - 3R1):
     \[
     \begin{pmatrix}
     1 & 2 & 3 & 0 \\
     0 & 0 & -3 & 2 \\
     0 & -4 & -8 & 3 \\
     6 & 8 & 7 & 5
     \end{pmatrix}
     \]
   - Subtract 6 times Row 1 from Row 4 (R4 = R4 - 6R1):
     \[
     \begin{pmatrix}
     1 & 2 & 3 & 0 \\
     0 & 0 & -3 & 2 \\
     0 & -4 & -8 & 3 \\
     0 & -4 & -11 & 5
     \end{pmatrix}
     \]

2. **Make the second column's element below the second row zero**:
   - Subtract Row 3 from Row 4 (R4 = R4 - R3):
     \[
     \begin{pmatrix}
     1 & 2 & 3 & 0 \\
     0 & 0 & -3 & 2 \\
     0 & -4 & -8 & 3 \\
     0 & 0 & -3 & 2
     \end{pmatrix}
     \]

Now, the matrix is in upper triangular form.

### Step 2: Determine the Rank of Matrix \( A \)
The rank of a matrix is the number of non-zero rows in its row-echelon form (triangular form).

The matrix in its final triangular form is:
\[
\begin{pmatrix}
1 & 2 & 3 & 0 \\
0 & 0 & -3 & 2 \\
0 & -4 & -8 & 3 \\
0 & 0 & -3 & 2
\end{pmatrix}
\]

All four rows are non-zero, so the **rank of matrix \( A \)** is **4**.

Would you like more details or have any questions? Here are 5 related questions you might consider:

1. What happens if one of the rows becomes entirely zero during Gaussian elimination?
2. How can you determine if a matrix is invertible from its rank?
3. What are the differences between row-echelon form and reduced row-echelon form?
4. How do the elementary row operations affect the determinant of a matrix?
5. How can the rank of a matrix be used to solve systems of linear equations?

**Tip:** When performing row operations, keep track of each step carefully to avoid errors, as these can affect the final rank calculation.

Learn how to reduce matrix A to triangular form using Gaussian elimination and determine its rank. This problem involves linear algebra concepts such as Gaussian elimination and matrix operations. Suitable for advanced high school students and above.

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