Understand various matrix operations, matrix decompositions, factorization and related operations
1. The rank of A =
2 -3 1
1 -2 5
-4 6 -7 is:
2. The minor M(1,3) of A =
1 -2 5
-4 2 -3
-5 3 -2 is:
3. The cofactor C(2,1) of A =
3 -2 1
6 -1 -5
-2 4 -3 is:
4. The minor M(3,2) of A =
3 -2 -3
-4 1 6
1 -2 3 is:
5. Eigenvalues of A =
2 3
2 1 are:
6. Eigenvectors of A =
0 1
-2 -3 are:
7. Eigenvalues of a Hermitian matrix are:
8. The conjugate transpose of matrix A =
1+2j 5-2j -6+2j
-1-3j -1-5j -3-j
2+3j -2-4j 2+j is:
9. Let A be a real n×n matrix. If A is both orthogonal and symmetric, what must A be?
10. If A is a non-zero nilpotent matrix, what is true about its determinant and trace?
11. Let A be a square matrix such that A is similar to a diagonal matrix D. What can be said about the minimal polynomial of A?
12. Let A be an n×n matrix over ℂ such that Aⁿ = I. Which of the following must be true about the eigenvalues of A?
13. Suppose A is a real symmetric matrix and λ is an eigenvalue of A with eigenvector x. Which of the following is always true?
14. Let A be a diagonalizable matrix with a repeated eigenvalue λ. Which of the following is true?
15. Let A be an upper triangular matrix. What can be said about its eigenvalues?
16. Suppose A is an n×n matrix such that A is not invertible. Which of the following is necessarily true?
17. Let A be an n×n real matrix such that Aᵗ = A⁻¹. What type of matrix is A?
18. Let A be a non-diagonalizable n×n matrix. Which of the following must be true?
19. Let A be an n×n matrix such that A^k = 0 for some positive integer k. What can be said about the minimal polynomial of A?
20. Let V be a vector space of dimension n and let T: V → V be a linear operator such that T² = T. Which of the following is necessarily true?
21. Which condition is necessary for a square matrix A to be diagonalizable?
22. Let T: ℝⁿ → ℝⁿ be a linear operator such that T has no non-trivial invariant subspaces. Which of the following must be true?
23. Let A be a square matrix. If A is diagonalizable, which of the following is true?
24. Let A be a 4×4 real matrix such that A^T = -A. What can be said about the eigenvalues of A?
25. If A and B are n×n matrices such that AB = BA, which of the following is always true?
26. Let V be an inner product space and T: V → V be a linear operator such that ⟨T(x), y⟩ = ⟨x, T(y)⟩ for all x, y ∈ V. Which of the following must be true?
27. Suppose A is a real matrix such that A^T A = A A^T. Which of the following is necessarily true?
28. If A is a square matrix and A² = 0, which of the following must be true?