Given that `A=[(alpha, beta),(gamma, - alpha)] and A^2 =3 I ` then :

Options

A. `1+ alpha^2 + beta gamma = 0 `
B. `1- alpha^2 + beta gamma = 0 `
C. `3- alpha^2 - beta gamma = 0 `
D. `3 + alpha^2 + beta gamma = 0 `

Answer

To solve the problem, we need to find the relationship between the elements of the matrix \( A \) given that \( A^2 = 3I \), where \( I \) is the identity matrix.

Given:
\[ A = \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix} \]
and
\[ A^2 = 3I \]

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

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \cdot A = \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix} \cdot \begin{pmatrix} \alpha & \beta \\ \gamma & -\alpha \end{pmatrix} \]

### Step 2: Perform the multiplication

Calculating the elements of \( A^2 \):

1. First row, first column:
\[ \alpha \cdot \alpha + \beta \cdot \gamma = \alpha^2 + \beta\gamma \]

2. First row, second column:
\[ \alpha \cdot \beta + \beta \cdot (-\alpha) = \alpha\beta - \beta\alpha = 0 \]

3. Second row, first column:
\[ \gamma \cdot \alpha + (-\alpha) \cdot \gamma = \gamma\alpha - \alpha\gamma = 0 \]

4. Second row, second column:
\[ \gamma \cdot \beta + (-\alpha) \cdot (-\alpha) = \gamma\beta + \alpha^2 \]

Putting it all together, we have:
\[ A^2 = \begin{pmatrix} \alpha^2 + \beta\gamma & 0 \\ 0 & \gamma\beta + \alpha^2 \end{pmatrix} \]

### Step 3: Set \( A^2 \) equal to \( 3I \)

Since \( A^2 = 3I \), we can write:
\[ \begin{pmatrix} \alpha^2 + \beta\gamma & 0 \\ 0 & \gamma\beta + \alpha^2 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

### Step 4: Equate the corresponding elements

From the above equation, we can equate the elements:
1. \( \alpha^2 + \beta\gamma = 3 \)
2. \( \gamma\beta + \alpha^2 = 3 \)

Both equations give us the same relationship:
\[ \alpha^2 + \beta\gamma = 3 \]

### Step 5: Rearranging the equation

Rearranging the equation gives:
\[ 3 - \alpha^2 - \beta\gamma = 0 \]

### Conclusion

Thus, the correct option that matches our derived equation is:
\[ 3 - \alpha^2 - \beta\gamma = 0 \]

This corresponds to option 3.