Matrix Operations

1. Matrix Multiplication
   • Note that the inner dimensions need to match up
2. Transpose
   • $A_{ij}=A_{ji}^T$, this means swap the rows and the columns
3. Inverse
   • $A{A}^{-1}=I=A^{-1}A$
   • Not all matrices have an inverse
   • The following statements are equivalent
     • A is invertible
     • The rank of A is n
     • RREF of A is the identity matrix
     • For all $b\in {\mathbb{R}}^n$ the system $Ax=b$ has a unique solution
     • The columns of A are linearly independent
     • The column space of A is ${\mathbb{R}}^n$
   • The way you find the inverse depends on the size of the matrix but for a simple 2x2 matrix…
   • $A^{-1}=\frac{1}{(ad-bc)}\left[\begin{array}{ccc}d& & -b\\ -c& & a\end{array}\right]$

Some properties include…

1. $(AB{)}^T=B^T{A}^T$
2. $(A+B{)}^T=A^T+B^T$
3. $(tA{)}^{-1}=\frac{1}{t}{A}^{-1}$
4. $(AB{)}^{-1}=A^{-1}{B}^{-1}$