Consider the system of linear equations: x + y + z = 1 4x + 5y + z = 2 16x + 25y + ^2 z = 5 Let the system be inconsistent for exactly two values of , say ₁ and ₂ . If A = bmatrix ₁ & 1 0 & ₂ bmatrix , then the trace of the matrix A^3 is

Step-by-step solution

For the system to be inconsistent, the determinant of the coefficient matrix must be zero: = vmatrix 1 & 1 & 1 4 & 5 & 16 & 25 & ^2 vmatrix = 0 Applying C₂ C₂ - C₁ and C₃ C₃ - C₁ : = vmatrix 1 & 0 & 0 4 & 1 & - 4 16 & 9 & ^2 - 16 vmatrix = ( ^2 - 16) - 9( - 4) = 0 ( - 4)( + 4 - 9) = 0 ( - 4)( - 5) = 0 Thus, = 4 or = 5 . We must verify that these values make the system inconsistent. For = 4 , the equations are x+y+z=1 , 4x+5y+4z=2 , and 16x+25y+16z=5 . From the first two equations, y = -2 and x+z = 3 . Substituting