To find the row echelon form of the matrix \( A \) and determine when the rank is 3, we proceed by performing Gaussian elimination. We'll reduce the matrix to row echelon form and then analyze the conditions under which the rank is 3.

The matrix \( A \) is:

\[
A = \begin{pmatrix}
1 & 1 & -1 & 0 \\
4 & 4 & -3 & 1 \\
p & 2 & 2 & 2 \\
9 & 9 & p & 3
\end{pmatrix}
\]

### Step 1: Apply Row Operations
We will perform row operations to convert \( A \) into its row echelon form.

Let's begin the Gaussian elimination process.The row echelon form of the matrix \( A \) is:

\[
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

This indicates that the matrix \( A \) is already in row echelon form, and the rank is 4 for any value of \( p \). This seems to contradict our earlier analysis because we needed a rank of 3.

### Understanding the Discrepancy
The calculation of the determinant \( p^2 + 4p - 12 = 0 \) gave us the values \( p = -6 \) and \( p = 2 \), which indicated that the matrix could potentially drop to rank 3. However, the row echelon form suggests that the matrix \( A \) does not drop rank under normal circumstances.

The row echelon form we calculated assumes that the matrix does not have special values of \( p \). If we substitute \( p = -6 \) or \( p = 2 \), the row echelon form should reveal one of the rows becoming dependent, thus reducing the rank to 3. Let's check the row echelon form for these specific values of \( p \).It seems like I can’t do more advanced data analysis right now. Please try again later. 

You might want to manually substitute \( p = -6 \) and \( p = 2 \) into the matrix and then perform Gaussian elimination (row reduction) to see how the row echelon form changes. This should confirm whether the matrix rank reduces to 3 for those values of \( p \).

If you have any questions or would like to explore another approach, feel free to ask!

Explore how to find the row echelon form of matrix A and determine its rank for various values of parameter p. This problem involves using Gaussian elimination and understanding how matrix rank changes with different parameter values. Suitable for advanced undergraduate students and beyond.

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