[[Universal Quantifier|For Some]] [[Bijective]] [[Linear Transformation]] $T: \R^n \to \R^m$, the [[Inverse Function|Inverse Transformation]] of $T$ is equal to the [[Inverse Matrices|Inverse Matrix]] of the the corresponding [[Matrix]] $A$ . An [[Inverse Matrices|Inverse Matrix]] of some [[Matrix]] $A$ is the [[Inverse Value]] under [[Matrix Product|Matrix Products]]. $A$ is invertible if [[Existential Quantifier|there exists]] another matrix $A^{-1}$ such that both multiplied is equal to [[Identity Matrix]]. $\huge A A^{-1} = I$ $ \huge A^{-1}{A \vec x} = I \vec x = \vec x $ For some $n\times n$ [[Matrix]] $A$ to be [[Inverse Function|Invertible]], it must have a [[Rank]] of $n$, its [[Reduced Row Echelon Form]] be the [[Identity Matrix]], or it has a non zero [[Determinant]]. $ \exists A^{-1} \iff \op{rank}(A)=n \iff \op{rref}(A) = I_{n} \iff \det A \neq 0 $ #### Algebraic Properties The the inverse of the inverse of some [[Matrix]] $A$ is simply $A$ $ \huge A A^{-1} = A^{-1} A = I$ ###### Inverse of a transpose $\huge \pa{ A^{\intercal} }^{-1} = \pa{ A^{-1}}^{\intercal} $ ###### Scalar Distribution $\huge\begin{align*} \let k &\in \R \\ \let A &\in M_{n\times n} \\ \pa{kA}^{-1} &= \frac{1}{k}A^{-1} \end{align*}$ ###### Inverse Determinant $\huge \det\pa{ A^{-1} } = \frac{1}{\det A} $ >[!example] Young's ~~Conjecture~~ Fact $\huge \begin{align} ABA^{-1}B^{-1} = \text{nah fam} \end{align}$ ### Computing Inverses #### $2\times 2$ Method $\huge\begin{align*} A &= \mat{a&b\\c&d} \\ A^{-1} &= \frac{1}{\det A} \mat{ d&-b\\-c&a} \\ &= \frac{1}{ad-bc}\mat{d&-b\\c&a} \end{align*}$ #### Row Operation Method $\huge\begin{align*} A &= \mat{5&3\\1&1} \\ A\hat i &= \mat{5 \\ 1} \\ A^{-1} \hat i &= \vec v_1 \\ AA^{-1} \hat i &= A\vec{v_{1}}\\ \vec i &= A\vec v_{1} \\ &\implies \augmented{cc|c}{ 5&3&1\\ 1&1&0 } & \\ &\sim \mat{ 1&0& \frac{1}{2}\\0&1& -\frac{1}{2}} \\ \hat j &= A\vec v_{2}\\ &\implies \cdots \sim \augmented{cc|c}{1&0& \frac{-3}{2} \\ 0 & 1 & \frac{5}{2}} \end{align*}$ $\huge A^{-1} = \mat{\frac{1}{2} & \frac{-3}{2} \\ \frac{-1}{2} & \frac{5}{2}}$ $\huge \augmented{c|c}{A & I} \sim \augmented{c|c}{I & A^{-1} } $ $\huge \begin{align*} \augmented{c|c}{A & I} \sim \cdots \sim \augmented{ccc|c}{ \vdots & \vdots & \vdots \\ 0 & 0 & 0} \iff \det{A} &= 0 \end{align*}$ #### Cofactor Method $\huge\begin{align*} A &= \mat{ \partial_{1} & \partial_{12} & \partial_{13}\\ \partial_{21} & \partial_{22} & \partial_{23}\\ \partial_{31} & \partial_{32} & \partial_{33}\\ }\\ A^{-1} &= \frac{1}{\det A} \mat{ C_{1,1} & C_{2,1} & C_{3, 1} \\ C_{1,2} & C_{2,2} & C_{3, 2} \\ C_{1,3} & C_{2,3} & C_{3, 3} \\ } \end{align*}$ >[!tip] [[../../02 Areas/Math/Determinant#Computing#Cofactor Expansion|See here for computing $C_{i, j}$]]