12. Let

 Classify matrices A, B, C, and D according to the categories: (1) Positive Definite, (2) Negative Definite, (3) Positive Semi-Defnite, (4) Negative Semi-Defite. Which of the following gives the correct order of matrices from category (1) to (4)?
(A) BACD
(B) ADCB
(C) ADBC
(D) DABC

13. Which of the following statements is FALSE about the algebraic and geomet- ric multiplicities of the eigenvalues of A? (A) The algebraic multiplicity of the eigenvalue -1 is 1. (B) The algebraic multiplicity of the eigenvalue 2 is 2. (C) The geometric multiplicity of the eigenvalue -1 is 1. (D) The geometric multiplicity of the eigenvalue 2 is 2.

14. ,, Find the dimen-sione of W1∩W2 sion of WinWz. (A) dim(Wi∩Wz)=1 (B) dim(W1∩W2)=2 (C) dim(W1∩W2)=3 (D) dim(W1∩W2)=4

15.  . (A)(B)(C)(D)

16. Let T : be a function. Which of the following defines a linear transformation? (A) T(a1, a2) = (1, a1) (B) T(a1, a2) = (a1, a2) (C) T(a1,a2) = (a1 + 1, a2) (D) T(a1,a2) = (a1 cosθ-a2sinθ, a1 sinθ+a2cos θ),   for a fixed angle θ

17.  Which of the following statements is FALSE? (A) S is a basis for a vector space V if and only if S is a minimal spanning set.  (B) V is a finite-dimensional vector space (dim(V) ＜∞), and W is a subspace of V. If dim(V) = dim(W), then W = V. (C) Let V and V'be vector spaces having the same finite dimension, and let T:V→ V be a linear transformation. Then T is one-to-one if and only if range(T) = V'  (D) Let V and V' be vector spaces of dimensions n and m, respectively. A linear transformation T : V -+ is invertible if and only if m = n.  

18.  Which of the following matrices are unitary? will 10:13 2100 (A)(B)(C)(D)

19. Considering the following matrix: The eigenvalues of the matrix﹕  The eigenvalues of the matrix A are (A)-1(B)1(C)6(D)7(E)2

20.  Which of the following statements are correct? (A) A linear system with fewer equations than unknowns may have no solution. (B) Every linear system with the same number of equations as unknowns has a unique solution. (C) A linear system with coefficient matrix A has an infinite number of solutions if and only if A can be row-reduced to an echelon matrix that includes some column containing no pivot. (D) If [A|b) and [B|c] are row-equivalent partitioned matrices, the linear systems Ax = b and Bx = c have the same solution set. (E) A linear system with a square coefficient matrix A has a unique solution if and only if A is row equivalent to the identity matrix.

21.  Which of the following statements are NOT correct? (A) Let A be a real n x n matrix. If A2 is invertible, then and A3 are invertible. (B) If A and B are invertible, then so is A+B, and . (C) Let A be a real m x n matrix and B a real n x m matrix. Then trace(AB) = trace(BA).  (D) Let A be a real m x matrix and B a real n x m matrix. Then det(AB) = det(BA). (E) Let A and B be real n x n matrices. Then AB = BA.

22.  Which of the following statements are NOT correct? (A) Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable.  (B) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly independent. (C) If A is an eigenvalue of a linear operator T, then each vector in the eigenspace is an eigenvector of T.  (D) A linear operator 7 on a finite-dimensional vector space is diagonalizable if and only if the multiplicity of each eigenvalue A equals the dimension of the corresponding eigenspace .  (E) If A is diagonalizable, then   is also diagonalizable.

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