Find the inverse of `A=[{:(2,-6),(1,-2):}]`.

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & -6 \\ 1 & -2 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of 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 \( A \):
- \( a = 2 \)
- \( b = -6 \)
- \( c = 1 \)
- \( d = -2 \)

Calculating the determinant:
\[ \text{det}(A) = (2)(-2) - (1)(-6) = -4 + 6 = 2 \]

### Step 2: Find the Adjoint of 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 \( A \):
- \( d = -2 \)
- \( -b = 6 \) (since \( b = -6 \))
- \( -c = -1 \) (since \( c = 1 \))
- \( a = 2 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} -2 & 6 \\ -1 & 2 \end{pmatrix} \]

### Step 3: Calculate the Inverse of A
The inverse of the matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{2} \cdot \begin{pmatrix} -2 & 6 \\ -1 & 2 \end{pmatrix} \]

### Step 4: Multiply by the Scalar
Now we multiply each element of the adjoint matrix by \( \frac{1}{2} \):
\[ A^{-1} = \begin{pmatrix} \frac{-2}{2} & \frac{6}{2} \\ \frac{-1}{2} & \frac{2}{2} \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ -\frac{1}{2} & 1 \end{pmatrix} \]

### Final Result
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} -1 & 3 \\ -\frac{1}{2} & 1 \end{pmatrix} \]