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 values of α (alpha) and β (beta) based on the given matrices A and A².

1. **Define the matrices**:
We have:
\[ A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \]
and
\[ A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix} \]

2. **Calculate \( A^2 \)**:
To find \( A^2 \), we perform matrix multiplication:
\[ A^2 = A \cdot A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \cdot \begin{pmatrix} a & b \\ b & a \end{pmatrix} \]

The multiplication is done as follows:
- The element at the first row and first column:
\[ a \cdot a + b \cdot b = a^2 + b^2 \]
- The element at the first row and second column:
\[ a \cdot b + b \cdot a = ab + ab = 2ab \]
- The element at the second row and first column:
\[ b \cdot a + a \cdot b = ba + ab = 2ab \]
- The element at the second row and second column:
\[ b \cdot b + a \cdot a = b^2 + a^2 \]

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

3. **Set the matrices equal**:
Now we set \( A^2 \) equal to the given matrix:
\[ \begin{pmatrix} a^2 + b^2 & 2ab \\ 2ab & a^2 + b^2 \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix} \]

4. **Equate corresponding elements**:
From the equality of matrices, we can equate the corresponding elements:
- From the first row and first column:
\[ \alpha = a^2 + b^2 \]
- From the first row and second column:
\[ \beta = 2ab \]

5. **Final values**:
Thus, we conclude:
\[ \alpha = a^2 + b^2 \]
\[ \beta = 2ab \]