Let's break down the given information and calculations:

Matrix Representation:

You have the matrix $A$ representing the transformation $T$ as:

$A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$

Determinant Calculation:

To check if the transformation $T$ is invertible, you compute the determinant of the matrix $A$:

$\text{det}(A) = (2 \times 3) - (0 \times 0) = 6 \neq 0$

Since the determinant is non-zero ($\text{det}(A) = 6$), the matrix $A$ is invertible.

Inverse Matrix:

The inverse of a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by:

$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

For the given matrix $A$:

$A^{-1} = \frac{1}{6} \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{pmatrix}$

So, the correct inverse matrix should be:

$A^{-1} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{pmatrix} = \begin{pmatrix} 0.5 & 0 \\ 0 & 0.3333 \end{pmatrix}$

However, in your text, you mentioned the inverse as:

$A^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}^{-1}$

This part appears to be incorrect. The correct inverse matrix has already been provided above.

Conclusion:

The transformation $T$ is indeed invertible because the determinant of its matrix representation is non-zero, and the correct inverse matrix has been calculated.

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Do you want more details or have any questions?

Here are 5 related questions to consider:

1. How do you find the determinant of a 3x3 matrix?
2. What conditions make a matrix non-invertible?
3. How can you check if a matrix is invertible without computing the determinant?
4. What is the geometric interpretation of an invertible matrix?
5. Can a matrix with a determinant of zero have an inverse?

Tip: When working with larger matrices, using row reduction methods can simplify the process of finding inverses and determinants.