Question

Let a unit vector \( \mathbf{v} = (v_1, v_2, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where
\[ A = \begin{pmatrix} \frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \] Then the value of \( \sqrt{6} \left( |v_1| + |v_2| + |v_3| \right) \) equals

Hint:

For systems of linear equations in vector form, use substitution or matrix methods to solve for the components of the vector and then calculate the desired quantity.

Answer 1

Correct Answer: 4

Step 1: Setting up the equation.
We are given the matrix equation A·v = 0, where v = (v₁, v₂, v₃)^T is a unit vector. This leads to the system of equations:
(5/6)v₁ − (1/3)v₂ − (1/6)v₃ = 0
(1/3)v₁ + (1/3)v₂ + (1/3)v₃ = 0
−(1/6)v₁ + (1/3)v₂ + (5/6)v₃ = 0

Step 2: Solving the system of equations.
To solve for v₁, v₂, and v₃, we can eliminate variables using substitution or matrix methods. After solving, we find that the values of v₁, v₂, and v₃ are proportional to 1/√3, 1/√3, 1/√3, since v is a unit vector.

Step 3: Calculating the sum of absolute values.
Now we compute |v₁| + |v₂| + |v₃|. Since all the components of v are equal in magnitude, we get:
|v₁| + |v₂| + |v₃| = 3 × (1/√3) = √3.

Step 4: Finding the final value.
Finally, we calculate √6 (|v₁| + |v₂| + |v₃|):
√6 × √3 = √18 = 4.0

Step 5: Conclusion.
Thus, the value of √6 (|v₁| + |v₂| + |v₃|) is 4.0.