If `A=[[2, -2], [-2, 2]] and B=[[1, 1], [1, 1]]`, then

Options

A. `A^(-1)=B`
B. `B^(-1)` exists
C. `A^(-1)` does not exists
D. `A^(-1)` exists

Answer

To solve the problem, we need to analyze the matrices \( A \) and \( B \) given as follows:

\[ A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \]

We need to determine the existence of the inverses of these matrices.

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is calculated using the formula:

\[ \text{det}(A) = ad - bc \]

For matrix \( A \):

- \( a = 2 \)
- \( b = -2 \)
- \( c = -2 \)
- \( d = 2 \)

Calculating the determinant:

\[ \text{det}(A) = (2)(2) - (-2)(-2) = 4 - 4 = 0 \]

### Step 2: Determine if \( A^{-1} \) exists

Since the determinant of \( A \) is \( 0 \), the inverse of matrix \( A \) does not exist.

### Step 3: Calculate the determinant of matrix \( B \)

Now, we calculate the determinant of matrix \( B \):

For matrix \( B \):

- \( a = 1 \)
- \( b = 1 \)
- \( c = 1 \)
- \( d = 1 \)

Calculating the determinant:

\[ \text{det}(B) = (1)(1) - (1)(1) = 1 - 1 = 0 \]

### Step 4: Determine if \( B^{-1} \) exists

Since the determinant of \( B \) is also \( 0 \), the inverse of matrix \( B \) does not exist.

### Conclusion

1. \( A^{-1} \) does not exist because \( \text{det}(A) = 0 \).
2. \( B^{-1} \) does not exist because \( \text{det}(B) = 0 \).

Thus, the correct option is that \( A^{-1} \) does not exist.