Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay, then a^2 + b^2 + c^2 + 2abc is equal to
Step-by-Step Solution
Key Concept: The system of equations is homogeneous: x - cy - bz = 0, -cx + y - az = 0, -bx - ay + z = 0. For a non-trivial solution (x, y, z) to exist, the determinant of the coefficient matrix must be zero.
The given system of equations is: x - cy - bz = 0, -cx + y - az = 0, -bx - ay + z = 0. Since there exists a non-trivial solution (x, y, z), the determinant of the coefficient matrix must be zero: |1 -c -b; -c 1 -a; -b -a 1| = 0. Expanding the determinant: 1(1 - a^2) + c(-c - ab) - b(ac + b) = 0 => 1 - a^2 - c^2 - abc - abc - b^2 = 0 => 1 - (a^2 + b^2 + c^2 + 2abc) = 0 => a^2 + b^2 + c^2 + 2abc = 1.
Correct Answer: 4