Let `A = [{:(1,2),(3,4):}]=[a_(ij)]`, where i, j = 1, 2, If its inverse matrix is `[b_(ij)]`, what is `b_(22)` ?

Options

A. -2
B. 1
C. `(3)/(2)`
D. `-(1)/(2)`

Explanation

To find the element \( b_{22} \) of the inverse matrix \( B \) of the matrix \( A \), we can follow these steps:

### Step 1: Define the Matrix \( A \)
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]

### Step 2: Calculate the Determinant of \( A \)
The determinant \( \text{det}(A) \) of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
\[ \text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \]

### Step 3: Find the Adjoint of \( A \)
The adjoint of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
\[ \text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]

### Step 4: Calculate the Inverse of \( A \)
The inverse of a matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{4}{-2} & \frac{-2}{-2} \\ \frac{-3}{-2} & \frac{1}{-2} \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \]

### Step 5: Identify \( b_{22} \)
The element \( b_{22} \) corresponds to the element in the second row and second column of the inverse matrix \( A^{-1} \):
\[ b_{22} = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( b_{22} \) is:
\[ \boxed{-\frac{1}{2}} \]