The inverse of matrix `[[3, -10], [2, -7]]` is

Options

A. `-[[-7, -2], [10, 3]]`
B. `[[-7, -2], [10, 3]]`
C. `-[[-7, 10], [-2, 3]]`
D. `[[-7, 10], [-2, 3]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 3 & -10 \\ 2 & -7 \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 3 \)
- \( b = -10 \)
- \( c = 2 \)
- \( d = -7 \)

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = (3)(-7) - (-10)(2) = -21 + 20 = -1 \]

### Step 2: Find the Adjoint of Matrix A
To find the adjoint of a \( 2 \times 2 \) matrix, we swap the elements on the main diagonal and change the signs of the off-diagonal elements. The adjoint of matrix \( A \) is given by:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Substituting the values from matrix \( A \):
\[ \text{adj}(A) = \begin{bmatrix} -7 & 10 \\ -2 & 3 \end{bmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the determinant and the adjoint:
\[ A^{-1} = \frac{1}{-1} \cdot \begin{bmatrix} -7 & 10 \\ -2 & 3 \end{bmatrix} = -1 \cdot \begin{bmatrix} -7 & 10 \\ -2 & 3 \end{bmatrix} \]
This simplifies to:
\[ A^{-1} = \begin{bmatrix} 7 & -10 \\ 2 & -3 \end{bmatrix} \]

### Final Answer
Thus, the inverse of the matrix \( A = \begin{bmatrix} 3 & -10 \\ 2 & -7 \end{bmatrix} \) is:
\[ A^{-1} = \begin{bmatrix} 7 & -10 \\ 2 & -3 \end{bmatrix} \]

---