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