Understand various matrix operations, matrix decompositions, factorization and related operations
1. If matrix A = [[1, 2], [3, 4]], which of the following statements is true about its eigenvalues?
2. Let A = [[2, 0], [0, 3]]. Which matrix B commutes with A (i.e., AB = BA)?
3. Consider matrix A = [[0, 1], [-2, -3]]. What is the trace and determinant of A?
4. For a 3×3 matrix A, if its rank is 2, what can be inferred?
5. If a matrix A is orthogonal, which of the following is always true?
6. If matrix A has eigenvalues 1, 2, and 3, what is the determinant of A?
7. If A is a square matrix such that A^2 = I, what are the possible eigenvalues of A?
8. Which of the following matrices is **not** invertible?
9. If A is a 3×3 matrix and det(A) = 5, what is det(2A)?
10. Which statement is true for any symmetric real matrix?
11. If A is a 2×2 matrix and A is invertible, which of the following must also be invertible?
12. What is the rank of the matrix [[1, 2], [2, 4]]?
13. Which of the following matrices is symmetric?
14. What is the trace of the matrix [[2, 1], [3, 4]]?
15. Which transformation preserves the length of vectors?
16. If A is diagonalizable, then it must have...
17. Which of the following matrices is idempotent (A^2 = A)?
18. What does the nullity of a matrix refer to?
19. If matrix A is singular, what does it imply?
20. The product of a matrix and its transpose is always...