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The given matrix \( A \) is:

\[
A = \begin{pmatrix}
4 & 3 & 1 \\
-2 & -3 & 0 \\
3 & 2 & 1
\end{pmatrix}
\]

### 1. **Correct Description of Matrix \( A \)**
To determine the correct description of the matrix, we need to check whether \( A \) is invertible or singular (non-invertible). This can be done by calculating the determinant of the matrix.

#### Determinant of \( A \):

\[
\det(A) = 4 \begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} - 3 \begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} + 1 \begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix}
\]

Now, we calculate the minors:

\[
\begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} = (-3)(1) - (0)(2) = -3
\]

\[
\begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} = (-2)(1) - (0)(3) = -2
\]

\[
\begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix} = (-2)(2) - (-3)(3) = -4 + 9 = 5
\]

Thus,

\[
\det(A) = 4(-3) - 3(-2) + 1(5) = -12 + 6 + 5 = -1
\]

Since the determinant of \( A \) is non-zero (\( \det(A) = -1 \)), the matrix is **invertible**.

### 2. **Finding \( A^{-1} \)**

The inverse of a 3x3 matrix \( A \) can be found using the formula:

\[
A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)
\]

Since \( \det(A) = -1 \), the inverse of \( A \) will be:

\[
A^{-1} = -\text{Adj}(A)
\]

The adjugate (or adjoint) matrix, \( \text{Adj}(A) \), is found by calculating the cofactor matrix and then transposing it.

#### Step 1: Cofactor Matrix of \( A \)

The cofactor matrix \( C \) is calculated by taking the determinant of the 2x2 submatrices obtained by deleting the corresponding row and column for each element of \( A \).

For \( C_{11} \):

\[
\text{Cofactor of } A_{11} = \begin{vmatrix} -3 & 0 \\ 2 & 1 \end{vmatrix} = -3
\]

For \( C_{12} \):

\[
\text{Cofactor of } A_{12} = -\begin{vmatrix} -2 & 0 \\ 3 & 1 \end{vmatrix} = 2
\]

For \( C_{13} \):

\[
\text{Cofactor of } A_{13} = \begin{vmatrix} -2 & -3 \\ 3 & 2 \end{vmatrix} = 5
\]

Following the same process for the rest of the entries, we get the cofactor matrix:

\[
C = \begin{pmatrix}
-3 & 2 & 5 \\
0 & -1 & 4 \\
3 & -2 & -1
\end{pmatrix}
\]

#### Step 2: Transpose of Cofactor Matrix

The adjugate matrix \( \text{Adj}(A) \) is the transpose of the cofactor matrix:

\[
\text{Adj}(A) = \begin{pmatrix}
-3 & 0 & 3 \\
2 & -1 & -2 \\
5 & 4 & -1
\end{pmatrix}
\]

#### Step 3: Multiply by \( \frac{1}{\det(A)} \)

Since \( \det(A) = -1 \), we multiply the adjugate matrix by \( -1 \):

\[
A^{-1} = -\text{Adj}(A) = \begin{pmatrix}
3 & 0 & -3 \\
-2 & 1 & 2 \\
-5 & -4 & 1
\end{pmatrix}
\]

### Final Answer:

\[
A^{-1} = \begin{pmatrix}
3 & 0 & -3 \\
-2 & 1 & 2 \\
-5 & -4 & 1
\end{pmatrix}
\]

Would you like any further clarifications or details on the solution?

### Follow-up Questions:
1. What are the properties of invertible matrices in general?
2. How does the determinant relate to the invertibility of a matrix?
3. Can you find the inverse of a 2x2 matrix using the same method?
4. What happens if the determinant of a matrix is zero?
5. How is the adjugate matrix used in other applications besides finding the inverse?

**Tip:** When solving matrix inversion problems, always check if the determinant is non-zero before proceeding with finding the inverse!

Consider the following matrix A = [[4, 3, 1], [-2, -3, 0], [3, 2, 1]]. Choose the correct description of A. Find A^-1 if it exists.

Explore how to calculate the determinant and inverse of a 3x3 matrix A = [[4, 3, 1], [-2, -3, 0], [3, 2, 1]]. Learn the process of finding A^-1 using the cofactor matrix and adjugate method. This problem covers key concepts in linear algebra, suitable for university students.

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