The inverse of the matrix `[[1, 3, 3], [1, 4, 3], [1, 3, 4]]` is

Options

A. `[[-7, -3, -3], [-1, 1, 0], [-1, 0, 1]]`
B. `[[7, 3, 3], [-1, 1, 0], [0, 0, -1]]`
C. `[[7, -3, -3], [-1, 1, 0], [-1, 0, 1]]`
D. `[[-7, 3, 3], [-1, 1, 0], [1, 0, -1]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:
- \( a = 1, b = 3, c = 3 \)
- \( d = 1, e = 4, f = 3 \)
- \( g = 1, h = 3, i = 4 \)

Calculating the determinant:
\[ \text{det}(A) = 1(4 \cdot 4 - 3 \cdot 3) - 3(1 \cdot 4 - 3 \cdot 1) + 3(1 \cdot 3 - 4 \cdot 1) \]
\[ = 1(16 - 9) - 3(4 - 3) + 3(3 - 4) \]
\[ = 1(7) - 3(1) + 3(-1) \]
\[ = 7 - 3 - 3 = 1 \]

### Step 2: Check if the Determinant is Non-Zero
Since \( \text{det}(A) = 1 \) (which is not zero), the inverse of the matrix exists.

### Step 3: Calculate the Adjoint of Matrix A
The adjoint of a matrix is the transpose of its cofactor matrix. We will calculate the cofactor matrix first.

#### Cofactor Calculation:
1. **Cofactor \( C_{11} \)**:
\[ C_{11} = \text{det}\begin{bmatrix} 4 & 3 \\ 3 & 4 \end{bmatrix} = (4 \cdot 4 - 3 \cdot 3) = 16 - 9 = 7 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = -\text{det}\begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix} = -((1 \cdot 4 - 3 \cdot 1)) = -(4 - 3) = -1 \]

3. **Cofactor \( C_{13} \)**:
\[ C_{13} = \text{det}\begin{bmatrix} 1 & 4 \\ 1 & 3 \end{bmatrix} = (1 \cdot 3 - 4 \cdot 1) = 3 - 4 = -1 \]

4. **Cofactor \( C_{21} \)**:
\[ C_{21} = -\text{det}\begin{bmatrix} 3 & 3 \\ 3 & 4 \end{bmatrix} = -((3 \cdot 4 - 3 \cdot 3)) = -(12 - 9) = -3 \]

5. **Cofactor \( C_{22} \)**:
\[ C_{22} = \text{det}\begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix} = (1 \cdot 4 - 3 \cdot 1) = 4 - 3 = 1 \]

6. **Cofactor \( C_{23} \)**:
\[ C_{23} = -\text{det}\begin{bmatrix} 1 & 3 \\ 1 & 3 \end{bmatrix} = -((1 \cdot 3 - 3 \cdot 1)) = -(3 - 3) = 0 \]

7. **Cofactor \( C_{31} \)**:
\[ C_{31} = \text{det}\begin{bmatrix} 3 & 3 \\ 4 & 3 \end{bmatrix} = (3 \cdot 3 - 3 \cdot 4) = 9 - 12 = -3 \]

8. **Cofactor \( C_{32} \)**:
\[ C_{32} = -\text{det}\begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix} = -((1 \cdot 4 - 3 \cdot 1)) = -(4 - 3) = -1 \]

9. **Cofactor \( C_{33} \)**:
\[ C_{33} = \text{det}\begin{bmatrix} 1 & 4 \\ 1 & 3 \end{bmatrix} = (1 \cdot 3 - 4 \cdot 1) = 3 - 4 = -1 \]

#### Cofactor Matrix:
\[ C = \begin{bmatrix} 7 & -1 & -1 \\ -3 & 1 & 0 \\ -3 & -1 & -1 \end{bmatrix} \]

#### Adjoint Matrix:
Taking the transpose of the cofactor matrix:
\[ \text{adj}(A) = \begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & -1 \\ -1 & 0 & -1 \end{bmatrix} \]

### Step 4: Calculate the Inverse of Matrix A
Using the formula for the inverse:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Since \( \text{det}(A) = 1 \):
\[ A^{-1} = 1 \cdot \text{adj}(A) = \begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & -1 \\ -1 & 0 & -1 \end{bmatrix} \]

### Final Answer:
The inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & -1 \\ -1 & 0 & -1 \end{bmatrix} \]