Find the rank of square matrix of 4*4 order 
-1 2 3 -2
2 -5 1 2
3 8 5 2
5 -12 -1 6

To find the rank of a 4x4 matrix, we need to determine the number of linearly independent rows or columns. The matrix given is:

\[
\begin{pmatrix}
-1 & 2 & 3 & -2 \\
2 & -5 & 1 & 2 \\
3 & 8 & 5 & 2 \\
5 & -12 & -1 & 6
\end{pmatrix}
\]

We'll perform row operations to reduce the matrix to its row echelon form (REF) and determine the rank by counting the non-zero rows.

### Step 1: Start with the original matrix.

\[
\begin{pmatrix}
-1 & 2 & 3 & -2 \\
2 & -5 & 1 & 2 \\
3 & 8 & 5 & 2 \\
5 & -12 & -1 & 6
\end{pmatrix}
\]

### Step 2: Perform Row Operations

- **Row 2**: Add 2 times **Row 1** to **Row 2**.
- **Row 3**: Add 3 times **Row 1** to **Row 3**.
- **Row 4**: Add 5 times **Row 1** to **Row 4**.

The matrix becomes:

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

- **Row 3**: Subtract 14 times **Row 2** from **Row 3**.
- **Row 4**: Add 2 times **Row 2** to **Row 4**.

The matrix becomes:

\[
\begin{pmatrix}
-1 & 2 & 3 & -2 \\
0 & -1 & 7 & -2 \\
0 & 0 & -84 & 24 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Step 3: Simplify the Matrix

Divide **Row 3** by \(-84\):

\[
\begin{pmatrix}
-1 & 2 & 3 & -2 \\
0 & -1 & 7 & -2 \\
0 & 0 & 1 & -\frac{2}{7} \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### Conclusion

The matrix is now in row echelon form. The non-zero rows indicate the rank of the matrix. Since there are 3 non-zero rows, the **rank of the matrix is 3**.

Would you like more details on the steps, or do you have any other questions?

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Here are 5 related questions:
1. How does row echelon form help in determining the rank of a matrix?
2. What is the difference between rank and determinant in a square matrix?
3. How would the rank change if one row was a multiple of another?
4. Can you have a matrix with rank 4 if one column is a linear combination of others?
5. How does the concept of linear independence relate to the rank of a matrix?

**Tip:** In row operations, always aim to create leading 1s (pivots) to simplify calculations and spot linear dependencies easily.

Learn how to find the rank of a 4x4 matrix using row operations and row echelon form. Follow a detailed step-by-step solution for the matrix [-1, 2, 3, -2; 2, -5, 1, 2; 3, 8, 5, 2; 5, -12, -1, 6]. Understand the concept of matrix rank, row operations, and linear independence.

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