Linear Transform MCQ - 5 - Mathematics MCQ

Step-by-Step Evaluation

1. Statement a): A is non singular
   • Correct. A matrix is non-singular if its determinant is not zero.
   • Calculate the Determinant of A:
     • det(A) = (5)(3) - (-1)(1) = 15 + 1 = 16
     • Since det(A) = 16 ≠ 0, matrix A is non-singular.
2. Statement b): A is not diagonalizable
   • Correct. To determine diagonalizability, we need to find the eigenvalues and their corresponding eigenspaces.
   • Find the Eigenvalues of A:
     • The characteristic equation is given by det(A - λI) = 0:
       det([5-λ, -1; 1, 3-λ]) = (5-λ)(3-λ) - (-1)(1) = (15 - 5λ - 3λ + λ²) + 1 = λ² - 8λ + 16 = 0
     • Solving λ² - 8λ + 16 = 0:
       λ = [8 ± √(64 - 64)] / 2 = [8 ± 0] / 2 = 4
     • Thus, the only eigenvalue is λ = 4 with algebraic multiplicity 2.
   • Determine the Geometric Multiplicity:
     • Solve (A - 4I)v = 0:
       A - 4I = [1, -1; 1, -1]
     • The system reduces to:
       x - y = 0
       x - y = 0
     • This yields only one linearly independent eigenvector, indicating geometric multiplicity 1.
     • Since geometric multiplicity (1) < algebraic multiplicity (2), matrix A is not diagonalizable.
3. Statement c): (1, 1) is an eigenvector.
   • Correct. From the eigenvalue analysis, eigenvectors satisfy x = y.
   • Verification:
     • Let v = [1; 1]. Then, Av = [5*(-1)*1 + (-1)*1; 1*1 + 3*1] = [5*1 -1*1; 1*1 + 3*1] = [4; 4] = 4*[1; 1]
     • Thus, v is an eigenvector corresponding to λ = 4.
4. Statement d): It has an eigenspace of dimension 2.
   • Incorrect. From the eigenvalue analysis, the eigenspace corresponding to λ = 4 has geometric multiplicity 1.
   • Explanation:
     • An eigenspace of dimension 2 would require two linearly independent eigenvectors for the eigenvalue λ = 4. However, as shown earlier, there is only one eigenvector satisfying (A - 4I)v = 0.
     • Therefore, the eigenspace has dimension 1, not 2.

Conclusion

Based on the evaluation:

• Statements a), b), and c) are correct.
• Statement d) is not correct.

By the definition, if is skew symmetric matrix, then
A^T = –A

a_ii = –a_ii for diagonal elements
2a_ii = 0 => a_ii = 0