To find the rank of the matrix and convert it to the identity matrix form, let's perform row reduction (Gaussian elimination). The matrix is:

$A = \begin{pmatrix} 8 & 1 & 3 & 6 \\ 0 & 3 & 2 & 2 \\ -8 & -1 & -3 & 4 \end{pmatrix}$

Step 1: Convert the matrix to row echelon form (REF)

We aim to make the matrix triangular by performing elementary row operations.

Row operations:

1. Start with the matrix $A$.

   $A = \begin{pmatrix} 8 & 1 & 3 & 6 \\ 0 & 3 & 2 & 2 \\ -8 & -1 & -3 & 4 \end{pmatrix}$

2. Eliminate the element in the first column of the third row. We do this by adding the first row to the third row:
   $R_3 = R_3 + R_1$.

   $A = \begin{pmatrix} 8 & 1 & 3 & 6 \\ 0 & 3 & 2 & 2 \\ 0 & 0 & 0 & 10 \end{pmatrix}$

Step 2: Analyze the row echelon form

Now that the matrix is in row echelon form, observe that the third row has all zeros except for the last column. We can determine the rank of the matrix based on the number of non-zero rows.

Here, there are 3 non-zero rows, so the rank of the matrix is 3.

Final matrix:

$\begin{pmatrix} 8 & 1 & 3 & 6 \\ 0 & 3 & 2 & 2 \\ 0 & 0 & 0 & 10 \end{pmatrix}$

This matrix has three non-zero rows, confirming that the rank of the matrix is 3.

---

Let me know if you need further details or have any questions!

Here are 5 related questions:

1. How is rank related to the number of independent rows or columns in a matrix?
2. What are the implications of a matrix having full rank?
3. Can a matrix with more rows than columns have a rank equal to the number of columns?
4. What role does Gaussian elimination play in finding the rank?
5. How is the concept of rank used in solving systems of linear equations?

Tip: In Gaussian elimination, always start row operations by focusing on one column at a time to simplify the matrix.