Question

If \(\begin{bmatrix} \alpha & \beta \gamma & -\alpha \end{bmatrix}\) is to be the square root of the two-rowed unit matrix, then \(\alpha, \beta\) and \(\gamma\) should satisfy the relation

• A. \(1 + \alpha^2 + \beta\gamma = 0\)
• B. \(1 - \alpha^2 - \beta\gamma = 0\)
• C. \(1 - \alpha^2 + \beta\gamma = 0\)
• D. \(\alpha^2 + \beta\gamma - 1 = 0\)

Hint:

Matrix square roots require careful multiplication, especially when off-diagonals may not vanish automatically.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
If \(A\) is a square root of \(I\), then \(A^2 = I\).

Step 2: Detailed Explanation:
Let \(A = \begin{bmatrix} \alpha & \beta \gamma & -\alpha \end{bmatrix}\). Then \(A^2 = \begin{bmatrix} \alpha^2 + \beta\gamma & \alpha\beta - \beta\alpha \gamma\alpha - \alpha\gamma & \beta\gamma + \alpha^2 \end{bmatrix} = \begin{bmatrix} \alpha^2 + \beta\gamma & 0 0 & \alpha^2 + \beta\gamma \end{bmatrix}\). For \(A^2 = I\), we need \(\alpha^2 + \beta\gamma = 1\) and off-diagonals 0 which are already 0. So \(\alpha^2 + \beta\gamma - 1 = 0\), or \(1 - \alpha^2 - \beta\gamma = 0\).

Step 3: Final Answer:
\(1 - \alpha^2 - \beta\gamma = 0\), which corresponds to option (B).