Inverse of `[{:(2,0),(0,3):}]` is `[{:(1/2,0),(0,1/3):}]` True or False.

Answer

To determine whether the statement about the inverse of the matrix is true or false, we need to verify if the product of the given matrix and its proposed inverse results in the identity matrix.

### Step-by-Step Solution:

1. **Identify the given matrix (A)**:
\[ A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]

2. **Identify the proposed inverse matrix (A⁻¹)**:
\[ A^{-1} = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{pmatrix} \]

3. **Multiply matrix A by its proposed inverse A⁻¹**:
\[ A \cdot A^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \cdot \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{pmatrix} \]

4. **Perform the multiplication**:
- For the element in the first row and first column:
\[ 2 \cdot \frac{1}{2} + 0 \cdot 0 = 1 \]
- For the element in the first row and second column:
\[ 2 \cdot 0 + 0 \cdot \frac{1}{3} = 0 \]
- For the element in the second row and first column:
\[ 0 \cdot \frac{1}{2} + 3 \cdot 0 = 0 \]
- For the element in the second row and second column:
\[ 0 \cdot 0 + 3 \cdot \frac{1}{3} = 1 \]

5. **Combine the results into a matrix**:
\[ A \cdot A^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

6. **Identify the resulting matrix**:
The resulting matrix is the identity matrix \( I \):
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

7. **Conclusion**:
Since \( A \cdot A^{-1} = I \), the proposed inverse is indeed the correct inverse of the matrix A. Therefore, the statement is **True**.