# a) If A and B are invertible n x n matrices, then so is A +B. b) The rank of a 4 x 5 matrix may be 5. c) The rank of a 5 x 4 matrix may be 5. d) If S and A are invertible, then (S-'AS)-1 =s-'A-'S. e) If A and B are given n x n matrices, then there is a unique n x n matrix X satisfying (A +X)B if B invertible. f) It is possible that a system Ax = b has a unique solution for some b if A is a 4 x 5 matrix. g) If A is a 5 x 5 matrix such that the system Ax 0 has only the trivial solution, then the system Ax = b is consistent for every be R*. h) An n xn matrix of rank n is invertible. i) The system Ax = 0 has only the trivial solution if and only if there are no free variables. j) For any two n xn matrices A and B, (A + B)? = A? + 2AB + B. k) For a 4 x 4 matrix A, det(3A)-3det(A). Which of the statements above are true? (Submit the corresponding number without parentheses.) (1) b, c, d, f, g. (2) a, b, d, i, k. (3) a, c, d, e, h. (4) d, e, g, h, i. (5) b, c, d, g, h, j. (6) a, c, d, f, g, k. (7) a, b, d, h, j, k. (8) a, b, c, e, g, h, k. (9) a, c, d, e, h, i, j. (10) b, d, e, f, h, j, k.

Question

a) If A and B are invertible n  n matrices, then so is A + B.
b) The rank of a 4  5 matrix may be 5.
c) The rank of a 5  4 matrix may be 5.
d) If S and A are invertible, then (S?1AS)?1 = S?1A?1S.
e) If A and B are given n  n matrices, then there is a unique n  n matrix X satisfying (A+X)B = A
if B invertible.
f) It is possible that a system Ax = b has a unique solution for some b if A is a 4  5 matrix.
g) If A is a 5  5 matrix such that the system Ax = 0 has only the trivial solution, then the system
Ax = b is consistent for every b 2 R5.
h) An n  n matrix of rank n is invertible.
i) The system Ax = 0 has only the trivial solution if and only if there are no free variables.
j) For any two n  n matrices A and B, (A + B)2 = A2 + 2AB + B2.
k) For a 4  4 matrix A, det(3A)=3det(A).
Which of the statements above are true?