The inverse of the matrix `[[2, 0, -1], [5, 1, 0], [0, 1, 3]]` is

Options

A. `[[3, -1, 1], [5, -2, 2], [-15, 6, -5]]`
B. `[[-3, 1, -1], [-5, 2, -2], [15, -6, 3]]`
C. `[[3, -1, 1], [-15, 6, -5], [5,- 2, 2]]`
D. `[[-3, 1, -1], [15, -6, 5], [-5, 2, -2]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \), 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 given by the formula:

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

For our matrix \( A \):

- \( a = 2, b = 0, c = -1 \)
- \( d = 5, e = 1, f = 0 \)
- \( g = 0, h = 1, i = 3 \)

Calculating the determinant:

\[ \text{det}(A) = 2(1 \cdot 3 - 0 \cdot 1) - 0(5 \cdot 3 - 0 \cdot 0) + (-1)(5 \cdot 1 - 0 \cdot 0) \]

This simplifies to:

\[ \text{det}(A) = 2(3) - 0 + (-1)(5) = 6 - 5 = 1 \]

### Step 2: Verify the Determinant

Since \( \text{det}(A) = 1 \), which is not equal to 0, the inverse of matrix \( A \) 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.

#### Step 3.1: Calculate the Cofactors

The cofactor \( C_{ij} \) is calculated by taking the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column, multiplied by \( (-1)^{i+j} \).

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = \text{det} \begin{bmatrix} 1 & 0 \\ 1 & 3 \end{bmatrix} = (1)(3) - (0)(1) = 3 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = -\text{det} \begin{bmatrix} 5 & 0 \\ 0 & 3 \end{bmatrix} = -((5)(3) - (0)(0)) = -15 \]

3. **Cofactor \( C_{13} \)**:
\[ C_{13} = \text{det} \begin{bmatrix} 5 & 1 \\ 0 & 1 \end{bmatrix} = (5)(1) - (1)(0) = 5 \]

4. **Cofactor \( C_{21} \)**:
\[ C_{21} = -\text{det} \begin{bmatrix} 0 & -1 \\ 1 & 3 \end{bmatrix} = -((0)(3) - (-1)(1)) = -1 \]

5. **Cofactor \( C_{22} \)**:
\[ C_{22} = \text{det} \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} = (2)(3) - (-1)(0) = 6 \]

6. **Cofactor \( C_{23} \)**:
\[ C_{23} = -\text{det} \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} = -((2)(1) - (0)(0)) = -2 \]

7. **Cofactor \( C_{31} \)**:
\[ C_{31} = \text{det} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = (0)(0) - (-1)(1) = 1 \]

8. **Cofactor \( C_{32} \)**:
\[ C_{32} = -\text{det} \begin{bmatrix} 2 & -1 \\ 5 & 0 \end{bmatrix} = -((2)(0) - (-1)(5)) = -5 \]

9. **Cofactor \( C_{33} \)**:
\[ C_{33} = \text{det} \begin{bmatrix} 2 & 0 \\ 5 & 1 \end{bmatrix} = (2)(1) - (0)(5) = 2 \]

#### Step 3.2: Form the Cofactor Matrix

The cofactor matrix \( C \) is:

\[ C = \begin{bmatrix} 3 & -15 & 5 \\ -1 & 6 & -2 \\ 1 & -5 & 2 \end{bmatrix} \]

#### Step 3.3: Transpose the Cofactor Matrix

Now, we take the transpose of the cofactor matrix to get the adjoint:

\[ \text{adj}(A) = C^T = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix} \]

### 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 \( \text{det}(A) = 1 \):

\[ A^{-1} = 1 \cdot \text{adj}(A) = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix} \]

### Final Answer

The inverse of the matrix \( A \) is:

\[ A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix} \]