If the multiplicative group of `2times2` matrices of the form `[[a, a], [a, a]]`, for `ane0 and ainR`, then the inverse of `[[2, 2], [2, 2]]` is

Options

A. `[[(1)/(2), (1)/(2)], [(1)/(2), (1)/(2)]]`
B. `[[(1)/(4), (1)/(4)], [(1)/(4), (1)/(4)]]`
C. `[[(1)/(8), (1)/(8)], [(1)/(8), (1)/(8)]]`
D. does not exist

Answer

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

### Step 1: Calculate the Determinant of the Matrix
The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
\[ a = 2, \quad b = 2, \quad c = 2, \quad d = 2 \]
Thus, the determinant is:
\[ \text{det}(A) = (2)(2) - (2)(2) = 4 - 4 = 0 \]

### Step 2: Determine the Existence of the Inverse
The inverse of a matrix exists if and only if the determinant is non-zero. Since we found that:
\[ \text{det}(A) = 0 \]
This indicates that the matrix \( A \) does not have an inverse.

### Conclusion
Therefore, the inverse of the matrix \( \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} \) does not exist.

### Final Answer
The answer is that the inverse does not exist.

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