You seem to be asking for the value of $p$ for which the matrix:

$A = \begin{pmatrix} 3 & p & p \\ p & 3 & p \\ p & p & 3 \end{pmatrix}$

is processed using elementary operations or has some specific properties (such as finding $p$ when the matrix is singular or when some elementary row operations are applied). Let me assume you are interested in finding the value of $p$ for which this matrix is singular, which means that the determinant of the matrix is zero.

Finding the determinant of $A$:

$A = \begin{pmatrix} 3 & p & p \\ p & 3 & p \\ p & p & 3 \end{pmatrix}$

To find the determinant of a 3x3 matrix, use the formula:

$\text{det}(A) = 3 \begin{vmatrix} 3 & p \\ p & 3 \end{vmatrix} - p \begin{vmatrix} p & p \\ p & 3 \end{vmatrix} + p \begin{vmatrix} p & 3 \\ p & p \end{vmatrix}$

Let’s calculate it step by step. I'll compute this determinant and solve for $p$ so that the matrix is singular (i.e., $\text{det}(A) = 0$).The determinant of the matrix $A$ is:

$\text{det}(A) = 2p^3 - 9p^2 + 27$

To find when the matrix is singular, we set the determinant equal to zero:

$2p^3 - 9p^2 + 27 = 0$

Next, I'll solve this cubic equation for $p$.The values of $p$ that make the matrix singular are:

$p = -\frac{3}{2} \quad \text{or} \quad p = 3$

These are the values of $p$ for which the determinant of the matrix $A$ is zero, making the matrix singular.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:

1. What is the significance of a matrix being singular?
2. How do elementary row operations affect the determinant of a matrix?
3. Can the matrix $A$ have an inverse when $p = 3$?
4. How do the eigenvalues of a matrix relate to its determinant?
5. What are the implications of having repeated roots for a cubic equation?

Tip: For a matrix to be invertible, its determinant must be non-zero, meaning the matrix is non-singular.