If A = `[{:(1, 2),(-4 , - 1):}]` , then `A^(-1)` is

Options

A. `(1)/(7) [{:(-1, 2),(4, 1):}]`
B. `(1)/(7) [{:(1, 2),(4-, -1):}]`
C. `(1)/(7) [{:(-1,-2),(4, 1):}]`
D. Does not exist

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 2 \\ -4 & -1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A

The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

- \( a = 1 \)
- \( b = 2 \)
- \( c = -4 \)
- \( d = -1 \)

Now, substituting these values into the determinant formula:

\[ \text{det}(A) = (1)(-1) - (2)(-4) = -1 + 8 = 7 \]

### Step 2: Find the Adjoint of Matrix A

The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For our matrix \( A \):

- \( d = -1 \)
- \( -b = -2 \)
- \( -c = 4 \)
- \( a = 1 \)

Thus, the adjoint of \( A \) is:

\[ \text{adj}(A) = \begin{pmatrix} -1 & -2 \\ 4 & 1 \end{pmatrix} \]

### Step 3: Calculate the Inverse of Matrix A

The inverse of a matrix \( A \) is given by the formula:

\[ A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)} \]

Substituting the values we found:

\[ A^{-1} = \frac{1}{7} \begin{pmatrix} -1 & -2 \\ 4 & 1 \end{pmatrix} \]

This results in:

\[ A^{-1} = \begin{pmatrix} -\frac{1}{7} & -\frac{2}{7} \\ \frac{4}{7} & \frac{1}{7} \end{pmatrix} \]

### Final Answer

Thus, the inverse of matrix \( A \) is:

\[ A^{-1} = \begin{pmatrix} -\frac{1}{7} & -\frac{2}{7} \\ \frac{4}{7} & \frac{1}{7} \end{pmatrix} \]

---