The matrix representation of T*T* is:

A=(2003)*A*=(20​03​)

To check if T*T* is invertible, we compute the determinant:

det⁡(A)=2⋅3−0⋅0=6≠0det(*A*)=2⋅3−0⋅0=6=0

Since det⁡(A)≠0det(*A*)=0, the transformation is invertible. The inverse matrix is:

A−1=(120013)*A*−1=(21​0​031​​)

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.

Explore the concept of matrix invertibility through a detailed example. Learn how to calculate determinants and find inverse matrices, crucial for understanding linear transformations. Suitable for advanced high school students and beyond.

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