The Invertible Matrix Theorem: Let be a square matrix. Then the following
statements are equivalent.

(a)
 is an invertible matrix.

(b)
 is row equivalent to the identity matrix.

(c)
 has pivot positions.

(d)
 The equation has only the trivial solution.

(e)
 The columns of form a linearly independent set.

(f)
 The linear transformation is onetoone.

(g)
 The equation has at least one solution for each in .

(h)
 The columns of span .

(i)
 The linear transformation maps onto .

(j)
 There is an matrix such that .

(k)
 There is an matrix such that .

(l)
 is an invertible matrix.
Determine if the matrices below are invertible.
A square matrix that is row equivalent to .
Invertible Not Invertible
.
Invertible Not Invertible
.
Invertible Not Invertible
.
Invertible Not Invertible
. (Hint: is not in echelon form)
Invertible Not Invertible