find the rank of the matrix 2 1 3 5, 4 2 1 3, 8 4 7 13, 8 4 -3-1

To find the rank of the given matrix, we will perform elementary row operations to convert it into its row echelon form (REF) or reduced row echelon form (RREF). The rank of a matrix is the number of non-zero rows in its REF or RREF.

Given matrix:

\[
A = \begin{pmatrix}
2 & 1 & 3 & 5 \\
4 & 2 & 1 & 3 \\
8 & 4 & 7 & 13 \\
8 & 4 & -3 & -1
\end{pmatrix}
\]

### Step 1: Convert to Row Echelon Form
We will perform row operations to simplify the matrix.

#### 1. Row 2: Subtract \(2 \times \text{Row 1}\) from Row 2:
\[
R_2 \rightarrow R_2 - 2R_1
\]
\[
R_2 = (4, 2, 1, 3) - 2(2, 1, 3, 5) = (4 - 4, 2 - 2, 1 - 6, 3 - 10) = (0, 0, -5, -7)
\]

#### 2. Row 3: Subtract \(4 \times \text{Row 1}\) from Row 3:
\[
R_3 \rightarrow R_3 - 4R_1
\]
\[
R_3 = (8, 4, 7, 13) - 4(2, 1, 3, 5) = (8 - 8, 4 - 4, 7 - 12, 13 - 20) = (0, 0, -5, -7)
\]

#### 3. Row 4: Subtract \(4 \times \text{Row 1}\) from Row 4:
\[
R_4 \rightarrow R_4 - 4R_1
\]
\[
R_4 = (8, 4, -3, -1) - 4(2, 1, 3, 5) = (8 - 8, 4 - 4, -3 - 12, -1 - 20) = (0, 0, -15, -21)
\]

At this stage, the matrix looks like:

\[
A = \begin{pmatrix}
2 & 1 & 3 & 5 \\
0 & 0 & -5 & -7 \\
0 & 0 & -5 & -7 \\
0 & 0 & -15 & -21
\end{pmatrix}
\]

#### 4. Row 4: Subtract \(3 \times \text{Row 2}\) from Row 4:
\[
R_4 \rightarrow R_4 - 3R_2
\]
\[
R_4 = (0, 0, -15, -21) - 3(0, 0, -5, -7) = (0, 0, -15 + 15, -21 + 21) = (0, 0, 0, 0)
\]

Now, the matrix becomes:

\[
A = \begin{pmatrix}
2 & 1 & 3 & 5 \\
0 & 0 & -5 & -7 \\
0 & 0 & -5 & -7 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Step 2: Identify the Rank
The row echelon form of the matrix has two non-zero rows. Therefore, the rank of the matrix is **2**.

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Would you like more details or have any further questions?

Here are 5 questions to deepen your understanding:
1. What is the relationship between the rank of a matrix and its determinant?
2. How can row reduction be used to determine the inverse of a matrix?
3. Can a matrix with more rows than columns have full rank?
4. What is the significance of a zero row in determining the rank?
5. How does the rank of a matrix relate to the solution of a system of linear equations?

**Tip:** A matrix's rank is always less than or equal to the smallest of its row or column dimensions.

Learn how to find the rank of a 4x4 matrix using elementary row operations to convert it into Row Echelon Form (REF). This detailed step-by-step guide explains each operation and shows that the rank of the matrix is 2. Ideal for students in Grades 10-12 studying linear algebra.

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