Find rank of matrix 1 2 3 4,-2-3 1 2,-3-4 5 8,1 3 10 14

We are tasked with finding the rank of the given matrix:

\[
\begin{pmatrix}
1 & 2 & 3 & 4 \\
-2 & -3 & 1 & 2 \\
-3 & -4 & 5 & 8 \\
1 & 3 & 10 & 14
\end{pmatrix}
\]

### Step 1: Form the augmented matrix

The matrix is:

\[
A = \begin{pmatrix}
1 & 2 & 3 & 4 \\
-2 & -3 & 1 & 2 \\
-3 & -4 & 5 & 8 \\
1 & 3 & 10 & 14
\end{pmatrix}
\]

### Step 2: Perform Gaussian elimination

We aim to transform the matrix into row echelon form (REF) by performing row operations.

- **Row 2 (R2)**: Add \( 2 \times R1 \) to R2:
  \[
  R2 = (-2 + 2, -3 + 4, 1 + 6, 2 + 8) = (0, 1, 7, 10)
  \]
  
- **Row 3 (R3)**: Add \( 3 \times R1 \) to R3:
  \[
  R3 = (-3 + 3, -4 + 6, 5 + 9, 8 + 12) = (0, 2, 14, 20)
  \]

- **Row 4 (R4)**: Subtract \( 1 \times R1 \) from R4:
  \[
  R4 = (1 - 1, 3 - 2, 10 - 3, 14 - 4) = (0, 1, 7, 10)
  \]

The resulting matrix is:

\[
\begin{pmatrix}
1 & 2 & 3 & 4 \\
0 & 1 & 7 & 10 \\
0 & 2 & 14 & 20 \\
0 & 1 & 7 & 10
\end{pmatrix}
\]

### Step 3: Eliminate entries below the pivot in column 2

- **Row 3 (R3)**: Subtract \( 2 \times R2 \) from R3:
  \[
  R3 = (0 - 0, 2 - 2, 14 - 14, 20 - 20) = (0, 0, 0, 0)
  \]

- **Row 4 (R4)**: Subtract \( 1 \times R2 \) from R4:
  \[
  R4 = (0 - 0, 1 - 1, 7 - 7, 10 - 10) = (0, 0, 0, 0)
  \]

The matrix is now:

\[
\begin{pmatrix}
1 & 2 & 3 & 4 \\
0 & 1 & 7 & 10 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Step 4: Determine the rank

The matrix is in row echelon form, and there are 2 non-zero rows. Therefore, the **rank of the matrix is 2**.

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Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you compute the rank of a matrix using determinants?
2. What is the significance of the rank in solving systems of linear equations?
3. How does Gaussian elimination relate to finding the null space of a matrix?
4. Can the rank of a matrix change under certain operations, like transposing?
5. How can the rank be used to check if a set of vectors is linearly independent?

**Tip**: Always aim to reduce matrices to row echelon form to simplify rank determination!

Find the rank of matrix 1 2 3 4, -2 -3 1 2, -3 -4 5 8, 1 3 10 14

Learn how to find the rank of a 4x4 matrix using Gaussian elimination. The matrix is reduced to row echelon form, and the number of non-zero rows determines the rank. This problem involves concepts from linear algebra and is suitable for advanced high school or undergraduate students.

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