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

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2
Q

b)

A

A is row equivalent to the nxn identity matrix

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3
Q

c)

A

A has n pivot positions

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4
Q

d)

A

The equation Ax=0 has only the trivial solution

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5
Q

e)

A

The columns of A form a linearly independent set

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6
Q

f)

A

The linear transformation x -> Ax is one-to-one

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7
Q

g)

A

The equation Ax=b at least one solution for each b on Rn.

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8
Q

h)

A

The columns of A span Rn

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9
Q

i)

A

The linear transformation x->Ax maps Rn onto Rn

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10
Q

j)

A

There is an nxn matrix C such that CA = I

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11
Q

k)

A

There is an nxn matrix such that AD = I

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12
Q

m)

A

m) The columns of A form a basis of Rn

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13
Q

n)

A

n) Col A = Rn

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14
Q

o)

A

o) dim Col A = n

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15
Q

p)

A

p) rank A = n

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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