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 one-to-one.
-
(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