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

Options

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

Answer

To solve the problem, we need to find the relationship between the elements of the matrix \( A \) and the elements of the matrix \( A^2 \).

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

### 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 multiplication

Using the rule of matrix multiplication (the dot product of rows and columns), we calculate each element of \( A^2 \):

1. **Element (1,1)**:
\[ a \cdot a + b \cdot b = a^2 + b^2 \]

2. **Element (1,2)**:
\[ a \cdot b + b \cdot a = ab + ba = 2ab \]

3. **Element (2,1)**:
\[ b \cdot a + a \cdot b = ba + ab = 2ab \]

4. **Element (2,2)**:
\[ b \cdot b + a \cdot a = b^2 + a^2 \]

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

### Step 3: Set the matrices equal to each other

We know that:
\[ A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix} \]

So we can equate the corresponding elements:
1. From the (1,1) position:
\[ \alpha = a^2 + b^2 \]

2. From the (1,2) position:
\[ \beta = 2ab \]

### Conclusion

We have derived the relationships:
\[ \alpha = a^2 + b^2 \quad \text{and} \quad \beta = 2ab \]

### Final Answer

Thus, the correct relationships are:
- \( \alpha = a^2 + b^2 \)
- \( \beta = 2ab \)