The statement in the problem is true: If $A \times B = I$, then $B$ is the inverse of matrix $A$. This is the definition of an inverse matrix.

Now, to confirm if matrix $D$ is the inverse of matrix $C$, we need to calculate the product $C \times D$ and check if it results in the identity matrix.

Given matrices: $C = \begin{bmatrix} 3 & 2 \\ 5 & 4 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & -1 \\ -2.5 & 1.5 \end{bmatrix}$

Let's perform the matrix multiplication $CD$:

$CD = \begin{bmatrix} 3 & 2 \\ 5 & 4 \end{bmatrix} \times \begin{bmatrix} 2 & -1 \\ -2.5 & 1.5 \end{bmatrix}$

To calculate the product:

1. First row, first column: $3(2) + 2(-2.5) = 6 - 5 = 1$
2. First row, second column: $3(-1) + 2(1.5) = -3 + 3 = 0$
3. Second row, first column: $5(2) + 4(-2.5) = 10 - 10 = 0$
4. Second row, second column: $5(-1) + 4(1.5) = -5 + 6 = 1$

Thus, the product $CD = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, which is the identity matrix. Therefore, matrix $D$ is indeed the inverse of matrix $C$.

Do you need further details or clarification on this calculation?

Here are some related questions you might find interesting:

1. How do you calculate the inverse of a 2x2 matrix?
2. What is the identity matrix, and why is it important?
3. How can we verify if two matrices are inverses of each other?
4. Can non-square matrices have an inverse?
5. How does matrix multiplication differ from scalar multiplication?

Tip: When working with matrices, always check the dimensions before multiplying to ensure the operation is valid!