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