Matrix Inverses

Multiplicative Identity

• The idea behind the multiplicative identity is that it returns the input as it is.

$$\begin{array}{r}x\times 1=x\end{array}$$

• For numbers 1 is the multiplicative identity.
• Similarly, for matrices $I$ is the multiplicative identity.

$$\begin{array}{r}AI=A\end{array}$$

Multiplicative Inverse

• The idea behind the inverse is can you find a corresponding number/matrix such that when multiplied with the original number/matrix, it returns the multiplicative identity.

$$\begin{array}{r}xy=1=yx\\ ⟹y\text{ is the inverse of x (and vice versa)}\\ \\ AB=I=BA\\ ⟹B\text{ is the inverse of }A\text{ and vice versa.}\end{array}$$

Constraints on shape of $A,B$

• Because $AB=I=BA$ both $A,B$ have to be square matrices of equal sizes.