If P is a `3xx3` matrix such that `P^T = 2P+I`, where `P^T` is the transpose of P and I is the `3xx3` identity matrix, then there exists a column matrix, `X = [[x],[y],[z]]!=[[0],[0],[0]]` such that

Options

A. `PX=[[0],[0],[0]]`
B. `PX = X`
C. `PX = 2X `
D. `PX =-X`

Explanation

To solve the problem, we start with the given equation involving the matrix \( P \):

### Step 1: Write down the given equation
We are given that:
\[ P^T = 2P + I \]
where \( P^T \) is the transpose of matrix \( P \) and \( I \) is the \( 3 \times 3 \) identity matrix.

**Hint:** Recall that the transpose of a matrix has specific properties, such as \( (AB)^T = B^T A^T \).

### Step 2: Take the transpose of both sides
Taking the transpose of both sides of the equation, we have:
\[ (P^T)^T = (2P + I)^T \]
Using the property of transposes, we get:
\[ P = 2P^T + I \]

**Hint:** Remember that the transpose of a sum is the sum of the transposes, and the transpose of a scalar multiplied by a matrix is the scalar multiplied by the transpose of the matrix.

### Step 3: Substitute \( P^T \) from the original equation
Now, we can substitute \( P^T \) from the original equation into this new equation:
\[ P = 2(2P + I) + I \]
This simplifies to:
\[ P = 4P + 2I + I \]
\[ P = 4P + 3I \]

**Hint:** Be careful with the distribution of the scalar when multiplying matrices.

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ P - 4P = 3I \]
\[ -3P = 3I \]
Dividing both sides by -3, we find:
\[ P = -I \]

**Hint:** When moving terms around, ensure you keep track of the signs.

### Step 5: Multiply \( P \) by a column matrix \( X \)
Let \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), where \( X \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \). Now we can compute \( PX \):
\[ PX = (-I)X \]
This simplifies to:
\[ PX = -X \]

**Hint:** The identity matrix \( I \) has the property that \( IX = X \).

### Conclusion
Thus, we have shown that:
\[ PX = -X \]
This means there exists a column matrix \( X \) such that \( PX = -X \).

The correct option is:
\[ \text{Option 4: } PX = -X \]