Invertible matrix theorem Flashcards
1
Q
a)
A
Let A be a square nxn matrix. Then the following statements are equivalent:
a) A is an invertible matrix
2
Q
b)
A
A is row equivalent to the nxn identity matrix
3
Q
c)
A
A has n pivot positions
4
Q
d)
A
The equation Ax=0 has only the trivial solution
5
Q
e)
A
The columns of A form a linearly independent set
6
Q
f)
A
The linear transformation x -> Ax is one-to-one
7
Q
g)
A
The equation Ax=b at least one solution for each b on Rn.
8
Q
h)
A
The columns of A span Rn
9
Q
i)
A
The linear transformation x->Ax maps Rn onto Rn
10
Q
j)
A
There is an nxn matrix C such that CA = I
11
Q
k)
A
There is an nxn matrix such that AD = I
12
Q
m)
A
m) The columns of A form a basis of Rn
13
Q
n)
A
n) Col A = Rn
14
Q
o)
A
o) dim Col A = n
15
Q
p)
A
p) rank A = n
16
Q
q)
A
q) Null A = {0}
17
Q
r)
A
r) dim Nul A = 0
18
Q
s)
A
The number 0 is not an eigenvalue of A
19
Q
t)
A
determinant of A is not zero
