Let A And B Be Invertible N × N Matrices C D Are A Generic N × N And N × P Matrices Respectively F = Dt D

Question

Let A and B be invertible n × n matrices. C, D are a generic n × n and n × p matrices respectively. F = Dt D.

Mark all statements that must be correct.

a). DtA is a p × p matrix

b). F is a p × p matrix

c). (CD) t = Ct Dt

d). (A B) −1 exists and is B−1 A−1

e). A B = B A

f). A C B is invertible.

g). B−1 A B is invertible.

h). F is invertible.

j). B−1A−1ACB = C

k). (At ) −1 exists and is (A−1 ) t

l). At = A (a.k.a. A is symmetric)

m). F t = F

n). v 0F ≥ 0, for any vector v of appropriate dimension

o). v 0F v ≥ 0, for any vector v of appropriate dimension (a.k.a. F is non-negative definite)

6. (14 pt) Let A and B be invertible n x n matrices. C, D are a generic n x n and n x p
matrices respectively. F = D D. Mark all statements that must be correct.
( ) D’A is a p x p matrix
( ) F is a p x p matrix
( ) (CD) = Ct Dt
( ) (AB) – exists and is B-1 A-1
() AB = BA
( ) ACB is invertible.
( ) B-A B is invertible.
() F is invertible.
( ) B-A -‘ACB = C
( ) (At) -1 exists and is (A-1)t
( ) At = A (a.k.a. A is symmetric)
( ) F = F
( ) v’F 2 0, for any vector v of appropriate dimension
( ) v’Fv 2 0, for any vector v of appropriate dimension (a.k.a. F is non-negative
definite)Math