The inverse of matrix `[[1, 3], [2, 7]]` is

Options

A. `[[7, -3], [-2, 1]]`
B. `[[-7, 3], [2, -1]]`
C. `[[7, -2], [-3, 1]]`
D. `[[-7, 2], [3, -1]]`

Answer

To find the inverse of the matrix \(\begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}\), we will follow these steps:

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

Calculating the determinant:
\[ \text{det}(A) = (1)(7) - (2)(3) = 7 - 6 = 1 \]

### Step 2: Calculate the Adjoint
The adjoint of a \(2 \times 2\) matrix is obtained by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal.

For the matrix \(A\):
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} 7 & -3 \\ -2 & 1 \end{bmatrix} \]

### Step 3: Calculate the Inverse
The inverse of a matrix \(A\) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{1} \cdot \begin{bmatrix} 7 & -3 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} 7 & -3 \\ -2 & 1 \end{bmatrix} \]

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

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