Dustin Payne

I am a Math major working towards being an actuary. I love the Royals, even though they usually aren't very good.

I also love the Invertible Matrix Theorem.

Let $A$ be a square $n\times n$ matrix. Then the following statements are equivalent. That is, for a given $A$, the statements are either all true or all false.

- a) $A$ an invertible matrix.
- b) $A$ is row equivalent to the $n\times n$ 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 $A\vec{x}→\vec{x}$ is one-to-one.
- g) The equation $A\vec{x}=\vec{b}$ has at least one solution for each $\vec{b}$ in $R^{n}$.
- h) The columns of $A$ span $R^{n}$.
- i) The linear transformation $A\vec{x}→\vec{x}$ maps $R^{n}$ onto $R^{n}$.
- j) There is an $n\times n$ matrix $C$ such that $CA=I$.
- k) There is an $n\times n$ matrix $D$ such that $AD=I$.
- l) $AT$ is an invertible matrix.
- m) The columns of $A$ form a basis of $R^{n}$.
- n) $ColA=R^{n}$
- o) dim$ColA=n$
- p) $rankA=n$
- q) $NulA={\vec{0}}$
- r) dim$NulA=0$
- s) The number 0 is not an eigenvalue of $A$ .
- t) The determinant of $A$ is not zero.