If \(P\) is a \(3 \times 3\) matrix such that \(P^{T}=2 P+I\), where \(P^{T}\) is the transpose of \(P\) and \(I\) is the \(3 \times 3\) identity matrix, then there exists a column matrix \(X=\left[\begin{array}{l}x \\\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\\ 0\end{array}\right]\) such that (A) \(P X=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]\) (B) \(P X=X\) (C) \(P X=2 X\) (D) \(P X=-X\)

The transpose of a matrix is obtained by swapping the rows and columns of the original matrix. In more formal terms, the element at the ith row and jth column of the original matrix becomes the element at the jth row and ith column of the transposed matrix. For example, if you have a matrix
\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
its transpose, denoted by \( A^T \), would be
\[ A^T = \begin{bmatrix} a & c \ b & d \end{bmatrix} \]
Understanding the properties of the transpose is crucial when solving matrix equations like the one given in the exercise. The transpose operation is fundamental to many areas of linear algebra and has special properties, such as \((AB)^T = B^T A^T\) when dealing with the product of two matrices.

The identity matrix, usually represented by \(I\), plays a role similar to the number 1 in matrix arithmetic. It is a square matrix with ones on the diagonal and zeros elsewhere. For any matrix \(A\) with dimensions that allow for multiplication, the products \(AI\) and \(IA\) are always equal to \(A\).

A \(3 \times 3\) identity matrix looks like:
\[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]
In our problem, the identity matrix serves to modify the value of matrix \(P\) in the equation \(P^T = 2P + I\). The addition of the identity matrix to \(2P\) effectively adds 1 to each diagonal element of \(P\) without changing the other elements, impacting the subsequent multiplication with \(X\).

Matrix multiplication is a way to combine two matrices to form a new matrix. Unlike element-by-element multiplication, matrix multiplication combines rows and columns using a summation. To multiply an \(m \times n\) matrix by an \(n \times p\) matrix, you sum the products of the corresponding elements of the rows of the first matrix and the columns of the second matrix.

The result is an \(m \times p\) matrix where each element comes from these summed products. This operation is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.

For instance:
\[ if\ P = \begin{bmatrix} p_{11} & p_{12} \ p_{21} & p_{22} \end{bmatrix} \text{ and } X = \begin{bmatrix} x_{1} \ x_{2} \end{bmatrix}, \text{ then } PX = \begin{bmatrix} p_{11}x_{1} + p_{12}x_{2} \ p_{21}x_{1} + p_{22}x_{2} \end{bmatrix} \]
In our exercise, understanding matrix multiplication is essential to decipher what \(PX\) would result in when applied to the initial matrix equation \(P^T = 2P + I\). By examining the multiplication rules and how they apply to the given matrices, it becomes clear why the correct choice is (C) \(PX = 2X\).