If `A=[(a,b),(b,a)]` and `A^(2)=[(alpha, beta),(beta, alpha)]`then `(alpha, beta)` is

Options

A. `(a^(2)+b^(2),ab)`
B. `(a^(2)+b^(2),2ab)`
C. `(a^(2)+b^(2),a^(2)-b^(2))`
D. `(2ab,a^(2)+b^(2))`

Answer

To solve the problem, we need to compute the square of the matrix \( A \) and then compare it with the given matrix \( A^2 \).

Given:
\[ A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \]
and
\[ A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix} \]

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:
\[ A^2 = A \cdot A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \cdot \begin{pmatrix} a & b \\ b & a \end{pmatrix} \]

### Step 2: Perform the matrix multiplication

The multiplication of two matrices is done as follows:

1. **First row, first column**:
\[ a \cdot a + b \cdot b = a^2 + b^2 \]

2. **First row, second column**:
\[ a \cdot b + b \cdot a = ab + ba = 2ab \]

3. **Second row, first column**:
\[ b \cdot a + a \cdot b = ba + ab = 2ab \]

4. **Second row, second column**:
\[ b \cdot b + a \cdot a = b^2 + a^2 \]

Putting these results together, we have:
\[ A^2 = \begin{pmatrix} a^2 + b^2 & 2ab \\ 2ab & a^2 + b^2 \end{pmatrix} \]

### Step 3: Compare with the given matrix \( A^2 \)

We know that:
\[ A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix} \]
From our calculation, we can equate the corresponding elements:
- \( \alpha = a^2 + b^2 \)
- \( \beta = 2ab \)

### Conclusion

Thus, the values of \( \alpha \) and \( \beta \) are:
\[ (\alpha, \beta) = (a^2 + b^2, 2ab) \]