If `A^(-1)=(-1)/(2)[[1, -4], [-1, 2]]`, then A=

Options

A. `[[2, 4], [1, 1]]`
B. `[[1, -4], [-1, 2]]`
C. `[[2, 4], [-1, 1]]`
D. `[[1, 1], [4, 2]]`

Answer

To find the matrix \( A \) given that \( A^{-1} = -\frac{1}{2} \begin{pmatrix} 1 & -4 \\ -1 & 2 \end{pmatrix} \), we can follow these steps:

### Step 1: Write down the expression for \( A^{-1} \)

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

### Step 2: Multiply the matrix by -2 to find \( A \)

To find \( A \), we take the inverse of \( A^{-1} \). Since \( A^{-1} = -\frac{1}{2} B \) where \( B = \begin{pmatrix} 1 & -4 \\ -1 & 2 \end{pmatrix} \), we can express \( A \) as:
\[ A = -2 A^{-1} \]
Substituting \( A^{-1} \):
\[ A = -2 \left(-\frac{1}{2} \begin{pmatrix} 1 & -4 \\ -1 & 2 \end{pmatrix}\right) \]
\[ A = \begin{pmatrix} 1 & -4 \\ -1 & 2 \end{pmatrix} \]

### Step 3: Adjust the matrix to find the correct form of \( A \)

Now, we need to adjust the matrix \( A \) based on the properties of the adjoint. The adjoint of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Given that the adjoint of \( A \) is \( \begin{pmatrix} 1 & -4 \\ -1 & 2 \end{pmatrix} \), we can find \( A \) by swapping the diagonal elements and changing the signs of the off-diagonal elements:
\[ A = \begin{pmatrix} 2 & 4 \\ 1 & 1 \end{pmatrix} \]

### Step 4: Conclusion

Thus, the matrix \( A \) is:
\[ A = \begin{pmatrix} 2 & 4 \\ 1 & 1 \end{pmatrix} \]

### Final Answer

The correct option is the first one: \( \begin{pmatrix} 2 & 4 \\ 1 & 1 \end{pmatrix} \).

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