Question

If \(3A = \begin{bmatrix} 1 & 2 & 2 \\[0.3em] 2 & 1 & -2 \\[0.3em] a & 2 & b \end{bmatrix}\) and \(AA^T = I\), then\(\frac{a}{b} + \frac{b}{a} =\):

• A. \( -\frac{5}{2} \)
• B. \( \frac{13}{6} \)
• C. \( \frac{13}{6} \)
• D. \( \frac{5}{2} \)

Hint:

For matrix multiplication and properties, ensure that: - The matrix \( A \) satisfies the equation \( AA^T = I \), meaning that \( A \) is an orthogonal matrix. - Use properties of orthogonal matrices (rows are orthogonal and have magnitude 1) to solve for unknowns.

Answer 1

The Correct Option is D

Step 1: Solving for matrix \( A \). Given that \( 3A = \begin{bmatrix} 1 & 2 & 2
2 & 1 & -2
a & 2 & b \end{bmatrix} \), we solve for \( A \): \[ A = \frac{1}{3} \begin{bmatrix} 1 & 2 & 2
2 & 1 & -2
a & 2 & b \end{bmatrix}. \] Step 2: Use the condition \( AA^T = I \). We compute \( AA^T \) and set it equal to the identity matrix, which gives us the relationships between \( a \) and \( b \). Step 3: Solving for \( a \) and \( b \). From the equations, we find that \( a = -5 \) and \( b = 5 \). Step 4: Compute \( \frac{a}{b} + \frac{b}{a} \). \[ \frac{a}{b} + \frac{b}{a} = \frac{-5}{5} + \frac{5}{-5} = -1 + (-1) = -2. \]