AERO GATE TOPPER
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Linear Algebra
AERO GATE TOPPER
1. If matrix A has a rank of 2, what is the rank of the matrix formed by multiplying A with a 2x2 invertible matrix B?
2. Given the matrix A \[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \] find the rank of A.
3. For a matrix A with eigenvalues 2, -3, and 5, what is the determinant of A?
4. If a matrix A satisfies A² - A = 0, where A is a 2x2 matrix, how many distinct eigenvalues does A have?
5. Which of the following statements is true for an invertible matrix A?
6. Let \( A \) be a 3x3 orthogonal matrix with a determinant of -1. Which of the following is true about the eigenvalues of \( A \)?
7. What is the rank of the following matrix? \[ C = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 4 & 5 & 6 \end{pmatrix} \]
8. What is the characteristic polynomial of the matrix D = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} ?
9. Which of the following is true for any two orthogonal vectors \( u \) and \( v \) in an inner product space?
10. Which of the following is NOT a property of a linear transformation?
11. Which of the following statements about the null space of a matrix A is TRUE?
12. Which of the following matrices does NOT have an inverse?
13. Calculate the determinant of the matrix \[ F = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \]
14. What are the eigenvalues of the matrix G = \begin{pmatrix} 4 & 0 \\ 0 & 5 \end{pmatrix} ?
15. What is the purpose of the Gram-Schmidt process?
16. How do you calculate the characteristic polynomial of a matrix A?
17. What does the Cayley-Hamilton theorem state?
18. Find the number of non-zero singular values for the matrix \[ H = \begin{pmatrix} 1 & 2 \\ 0 & 0 \\ 3 & 4 \end{pmatrix} \].
19. What is the Jordan normal form of a matrix used for?
20. What does RREF stand for in the context of linear systems?
21. Which of the following correctly defines a vector space?
22. What is the purpose of the Gram-Schmidt process?
23. What characterizes a nilpotent matrix?
24. Which property defines an orthogonal matrix?
25. What does the spectral theorem state for symmetric matrices?
26. What does the rank-nullity theorem state?
27. How is an eigenvalue defined for a matrix A?
28. When is a matrix considered diagonalizable?
29. Which of the following is a property of a linear transformation?
30. How is the characteristic polynomial of a matrix A defined?
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