If `A=[(2,3),(4,6)]` , then `A^(-1)=`

Options

A. `[(1,2),(-(3)/(2),2)]`
B. `[(2,-3),(4,6)]`
C. `[(-2,4),(-3,6)]`
D. Does not exist

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix} \), 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 \( A \):
- \( a = 2 \)
- \( b = 3 \)
- \( c = 4 \)
- \( d = 6 \)

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = (2)(6) - (3)(4) = 12 - 12 = 0 \]

### Step 2: Check if the Determinant is Zero
Since the determinant of matrix \( A \) is 0, it indicates that the matrix is singular.

### Step 3: Conclusion about the Inverse
A matrix has an inverse if and only if its determinant is non-zero. Since we found that the determinant is zero, we conclude that the inverse of matrix \( A \) does not exist.

Thus, the final answer is:
\[ A^{-1} \text{ does not exist.} \]

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