📘 Question
If
\[
A =
\begin{bmatrix}
\alpha & \beta \\
\gamma & -\alpha
\end{bmatrix}
\]
and \(A^2 = I\), find the relation between \(\alpha, \beta, \gamma\).
✏️ Step-by-Step Solution
Step 1: Compute \(A^2\)
\[
A^2 =
\begin{bmatrix}
\alpha & \beta \\
\gamma & -\alpha
\end{bmatrix}
\begin{bmatrix}
\alpha & \beta \\
\gamma & -\alpha
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
\alpha^2 + \beta\gamma & \alpha\beta – \alpha\beta \\
\alpha\gamma – \alpha\gamma & \gamma\beta + \alpha^2
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
\alpha^2 + \beta\gamma & 0 \\
0 & \alpha^2 + \beta\gamma
\end{bmatrix}
\]
Step 2: Compare with identity matrix
\[
A^2 =
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\]
So,
\[
\alpha^2 + \beta\gamma = 1
\]
✅ Final Answer
\[
\boxed{\alpha^2 + \beta\gamma = 1}
\]
💡 Key Concept
Equate corresponding elements after multiplication. Here both diagonal elements must be 1, giving the required condition.