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The problem involves transforming a set of random variables \( X_1, X_2, X_3 \) (with given covariance matrix \( C_X \)) into a new set of random variables \( Y_1, Y_2, Y_3 \) using a transformation matrix \( T \). You are asked to find the covariance matrix of the new random variables.

### Given:
1. Covariance matrix \( C_X \):
   \[
   C_X = \begin{bmatrix}
   3.0 & 1.8 & 1.1 \\
   1.8 & 3.0 & 1.8 \\
   1.1 & 1.8 & 1.1
   \end{bmatrix}
   \]

2. Transformation matrix \( T \):
   \[
   T = \begin{bmatrix}
   4 & -1 & -2 \\
   2 & 2 & 1 \\
   -3 & -1 & 3
   \end{bmatrix}
   \]

### To Find:
The covariance matrix \( C_Y \) of the new random variables \( Y_1, Y_2, Y_3 \), where:
\[
C_Y = T C_X T^T
\]
where \( T^T \) is the transpose of the transformation matrix \( T \).

I'll now calculate the new covariance matrix \( C_Y \).The covariance matrix \( C_Y \) of the new random variables \( Y_1, Y_2, Y_3 \) is:

\[
C_Y = \begin{bmatrix}
30.6 & 17.6 & -23.4 \\
17.6 & 51.1 & -22.8 \\
-23.4 & -22.8 & 20.1
\end{bmatrix}
\]

This matrix represents the covariances of the transformed variables.

Do you want further details on how this result was derived or have any questions?

Here are 5 related questions for further understanding:
1. How is a covariance matrix interpreted in the context of random variables?
2. Why do we use transformation matrices on covariance matrices?
3. How would the covariance matrix change if the transformation matrix \( T \) were different?
4. What does a negative value in the covariance matrix imply about the relationship between two variables?
5. Can you generalize this approach to higher dimensions?

**Tip**: To ensure matrix multiplication is valid, remember that the number of columns in the first matrix must equal the number of rows in the second.

Convert the variables X1, X2, X3 to new random variables Y1, Y2, Y3 using a given transformation matrix T. Find the covariance matrix of the new random variables.

Learn how to transform the covariance matrix of random variables using a linear transformation matrix. This example involves calculating the new covariance matrix from a given transformation matrix and initial covariance matrix, demonstrating matrix multiplication in linear algebra.

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