The invertible matrix theorem

The Invertible Matrix Theorem
• only works for $n \times n$ square matrices
• If one is true, then they are all true
• If one is false, then they are all false
How to apply the Invertible Matrix Theorem
• Showing a Matrix is invertible
• Shortcuts to know certain statements

Showing a Matrix is invertible or not invertible
Is the following matrix invertible?
Is the following matrix invertible? Use as few calculations as possible.
Understanding the Theorem
Assume that $A$ $A$ is a square $n \times n$ $n \times n$ matrix. Determine if the following statements are true or false:
1. If $A$ is an invertible matrix, then the linear transformation $x$ → $Ax$ maps $\Bbb{R}^n$ onto $\Bbb{R}^n$ .
2. If there is an $n \times n$ matrix $C$ such that $CA=I$ , then there is an $n \times n$ matrix $D$ such that $AD=I$
3. If the equation $Ax=0$ has only the trivial solution, then $A$ is not invertible.
4. If the equation $Ax=0$ has a non-trivial solution, then $A$ has less than $n$ pivots.
Can a square matrix with two identical rows be invertible? Why or why not?
Let $A$ and $B$ be $n \times n$ matrix. Show that if $AB$ is invertible, so is $B$ .