To find the inverse of the matrix \(\begin{pmatrix} 3 & 0 & 2 \\ 1 & 4 & 0 \\ 0 & 2 & 1 \end{pmatrix}\), we can follow these steps:

### 1. Find the determinant of the matrix
The determinant \(\text{det}(A)\) of a \(3 \times 3\) matrix \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\) is calculated as:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For the given matrix \(A = \begin{pmatrix} 3 & 0 & 2 \\ 1 & 4 & 0 \\ 0 & 2 & 1 \end{pmatrix}\), let's calculate:

\[
\text{det}(A) = 3(4 \cdot 1 - 0 \cdot 2) - 0(1 \cdot 1 - 0 \cdot 2) + 2(1 \cdot 2 - 4 \cdot 0)
\]
\[
\text{det}(A) = 3(4) - 0 + 2(2)
\]
\[
\text{det}(A) = 12 + 4 = 16
\]

### 2. Find the matrix of minors
The matrix of minors is calculated by finding the determinant of the \(2 \times 2\) submatrix that remains after removing the row and column of each element.

\[
M = \begin{pmatrix} 
\text{det}\begin{pmatrix} 4 & 0 \\ 2 & 1 \end{pmatrix} & \text{det}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} & \text{det}\begin{pmatrix} 1 & 4 \\ 0 & 2 \end{pmatrix} \\ 
\text{det}\begin{pmatrix} 0 & 2 \\ 2 & 1 \end{pmatrix} & \text{det}\begin{pmatrix} 3 & 2 \\ 0 & 1 \end{pmatrix} & \text{det}\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \\ 
\text{det}\begin{pmatrix} 0 & 2 \\ 4 & 0 \end{pmatrix} & \text{det}\begin{pmatrix} 3 & 2 \\ 1 & 0 \end{pmatrix} & \text{det}\begin{pmatrix} 3 & 0 \\ 1 & 4 \end{pmatrix}
\end{pmatrix}
\]

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

\[
M = \begin{pmatrix} 
4 & 1 & 2 \\ 
-4 & 3 & 6 \\ 
-8 & -2 & 12 
\end{pmatrix}
\]

### 3. Find the matrix of cofactors
To find the matrix of cofactors, apply the sign pattern to the matrix of minors:

\[
C = \begin{pmatrix} 
+M_{11} & -M_{12} & +M_{13} \\ 
-M_{21} & +M_{22} & -M_{23} \\ 
+M_{31} & -M_{32} & +M_{33} 
\end{pmatrix}
\]

\[
C = \begin{pmatrix} 
4 & -1 & 2 \\ 
4 & 3 & -6 \\ 
-8 & 2 & 12 
\end{pmatrix}
\]

### 4. Transpose the matrix of cofactors to get the adjugate matrix
Transpose the matrix \(C\) to get the adjugate matrix \(C^T\):

\[
C^T = \begin{pmatrix} 
4 & 4 & -8 \\ 
-1 & 3 & 2 \\ 
2 & -6 & 12 
\end{pmatrix}
\]

### 5. Divide the adjugate matrix by the determinant to get the inverse
Finally, divide each element of the adjugate matrix by the determinant:

\[
A^{-1} = \frac{1}{\text{det}(A)} C^T = \frac{1}{16} \begin{pmatrix} 
4 & 4 & -8 \\ 
-1 & 3 & 2 \\ 
2 & -6 & 12 
\end{pmatrix}
\]

\[
A^{-1} = \begin{pmatrix} 
\frac{4}{16} & \frac{4}{16} & \frac{-8}{16} \\ 
\frac{-1}{16} & \frac{3}{16} & \frac{2}{16} \\ 
\frac{2}{16} & \frac{-6}{16} & \frac{12}{16} 
\end{pmatrix}
\]

\[
A^{-1} = \begin{pmatrix} 
\frac{1}{4} & \frac{1}{4} & -\frac{1}{2} \\ 
-\frac{1}{16} & \frac{3}{16} & \frac{1}{8} \\ 
\frac{1}{8} & -\frac{3}{8} & \frac{3}{4} 
\end{pmatrix}
\]

So, the inverse of the matrix is:

\[
A^{-1} = \begin{pmatrix} 
0.25 & 0.25 & -0.5 \\ 
-0.0625 & 0.1875 & 0.125 \\ 
0.125 & -0.375 & 0.75 
\end{pmatrix}
\]

Would you like more details or have any questions?

### Follow-Up Questions:
1. What is the significance of the determinant in finding the inverse of a matrix?
2. How does the sign pattern work when finding the cofactor matrix?
3. Can all matrices be inverted? If not, under what conditions does a matrix not have an inverse?
4. How would you check if the computed inverse is correct?
5. What are some real-world applications of matrix inversion?

### Tip:
Always check the determinant before attempting to find the inverse of a matrix. If the determinant is zero, the matrix does not have an inverse.

Learn how to find the inverse of the matrix 3 0 2, 1 4 0, 0 2 1 with a step-by-step detailed solution. This problem involves calculating the determinant, matrix of minors, matrix of cofactors, adjugate matrix, and finally the inverse of a 3x3 matrix. Suitable for advanced high school students and beyond.

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