Question

If

\[ \begin{pmatrix} \frac{\sqrt{5}}{3} & -\frac{2}{3} & c \\ \frac{2}{3} & \frac{\sqrt{5}}{3} & d \\ a & b & 1 \end{pmatrix} \]

is a real orthogonal matrix, then \( a^2 + b^2 + c^2 + d^2 \) equals ...............

Hint:

For an orthogonal matrix, the sum of squares of each row (or column) must equal 1, and the dot product of any two distinct rows (or columns) must be zero. This is a direct consequence of the definition of an orthogonal matrix.

Answer 1

Correct Answer: 0

Step 1: Properties of an orthogonal matrix.
For a matrix to be orthogonal, its rows (and columns) must be orthonormal. This means that the dot product of any two distinct rows (or columns) must be zero, and the dot product of a row (or column) with itself must be 1. That is: \[ \text{Row 1} \cdot \text{Row 1} = 1, \quad \text{Row 2} \cdot \text{Row 2} = 1, \quad \text{Row 3} \cdot \text{Row 3} = 1. \] Additionally, the dot product between different rows should be 0.
Step 2: Apply the orthonormality conditions.
From the given matrix, the first row is \( \left( \frac{\sqrt{5}}{3}, -\frac{2}{3}, c \right) \). The condition for the first row to be a unit vector is: \[ \left( \frac{\sqrt{5}}{3} \right)^2 + \left( -\frac{2}{3} \right)^2 + c^2 = 1. \] This simplifies to: \[ \frac{5}{9} + \frac{4}{9} + c^2 = 1 \quad \Rightarrow \quad \frac{9}{9} + c^2 = 1 \quad \Rightarrow \quad c^2 = 0. \] Thus, \( c = 0 \).
Step 3: Apply the second row condition.
The second row is \( \left( \frac{2}{3}, \frac{\sqrt{5}}{3}, d \right) \). The condition for the second row to be a unit vector is: \[ \left( \frac{2}{3} \right)^2 + \left( \frac{\sqrt{5}}{3} \right)^2 + d^2 = 1. \] This simplifies to: \[ \frac{4}{9} + \frac{5}{9} + d^2 = 1 \quad \Rightarrow \quad \frac{9}{9} + d^2 = 1 \quad \Rightarrow \quad d^2 = 0. \] Thus, \( d = 0 \).
Step 4: Apply the third row condition.
The third row is \( (a, b, 1) \). The condition for the third row to be a unit vector is: \[ a^2 + b^2 + 1^2 = 1 \quad \Rightarrow \quad a^2 + b^2 + 1 = 1 \quad \Rightarrow \quad a^2 + b^2 = 0. \] Thus, \( a = 0 \) and \( b = 0 \).
Step 5: Conclusion.
We have \( a = 0 \), \( b = 0 \), \( c = 0 \), and \( d = 0 \), so the sum \( a^2 + b^2 + c^2 + d^2 = 0 \).
Final Answer: \[ \boxed{0}. \]