Question

If \( A = \begin{bmatrix} a & b \\ b & a \end{bmatrix} \) and \( (AI)^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix} \), where \( I \) is the identity matrix, then __________.

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

Hint:

Identity matrix does not change a matrix: \( AI = A \). Always simplify before squaring.

Answer 1

The Correct Option is A

Step 1: Use identity property.
Since \( I \) is identity matrix:
\[ AI = A \]

Step 2: Hence simplify expression.
\[ (AI)^2 = A^2 \]

Step 3: Write matrix \( A \).
\[ A = \begin{bmatrix} a & b b & a \end{bmatrix} \]

Step 4: Compute \( A^2 \).
\[ A^2 = \begin{bmatrix} a & b b & a \end{bmatrix} \begin{bmatrix} a & b b & a \end{bmatrix} \]

Step 5: Multiply matrices.
\[ A^2 = \begin{bmatrix} a^2 + b^2 & ab + ba ba + ab & b^2 + a^2 \end{bmatrix} \]
\[ = \begin{bmatrix} a^2 + b^2 & 2ab 2ab & a^2 + b^2 \end{bmatrix} \]

Step 6: Compare with given matrix.
\[ \begin{bmatrix} \alpha & \beta \beta & \alpha \end{bmatrix} \]
So,
\[ \alpha = a^2 + b^2,\quad \beta = 2ab \]

Step 7: Final Answer.
\[ \boxed{\alpha = a^2 + b^2,\; \beta = 2ab} \]