Inverse Matrix

inverse of a matrix

$BA=In$ $AB=In$

we call B the inverse of A $A^{-1}=$ theinverse of a

example

$$\begin{array}{r}A=\left[\begin{array}{cc}3& 1\\ 2& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}1& -1\\ -2& 3\end{array}\right]\end{array}$$

is B the inverse of A? take the cross product

if A × B = I, then B × A = I

Why do we care about inverses?

Solve

$$\begin{gathered} 3x+y=4 \\ 2x+y=2 \end{gathered}$$

$A\vec{x}=\vec{b}$

$$\left[\begin{array}{cc}3& 1\\ 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}4\\ 2\end{array}\right]$$

Multiply both sides by $A^{-1}$

$$\left[\begin{array}{cc}1& -1\\ -2& -3\end{array}\right]\left[\begin{array}{cc}3& 1\\ 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}1& -1\\ -2& -3\end{array}\right]\left[\begin{array}{c}4\\ 2\end{array}\right]$$ $$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ -2\end{array}\right]$$ $$\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ -2\end{array}\right]$$

THIS IS EFFECTIVELY WHAT WE DO IN BASIC ALGEBRA. IF WE HAVE $2x=3$ we can multiply each side by $\frac{1}{2}$ or the “inverse” of 2 like $\frac{1}{2}2x=\frac{1}{2}3$ yielding $1x=\frac{2}{3}$.

IF $A$ has an inverse than we can use this trick. if the matrix A isn’t square it doesn’t have an inverse. It has to be a square to have an inverse, or to be “invertable”

Not all square matrices are invertable. Only matrices where the [determinant] isn’t zero

if the matrix is invertible it only has one solution

to GET the inverse of a matrix we divide the [adjugate] by the [determinant] Hers how to find the formula for the https://en.wikipedia.org/wiki/Adjugate_matrix

MY PROFESSOR is teaching it a different way. Set up matrix [A | In] 3 1 | 1 0 2 1 | 0 1 Goal 1 0 | # # 0 1 | # #

if A is 2x2, just flip along the diagonal and change the signs on the other one, then divide by 1 over the determinant

OH thats the adjugate this is just for a 2x2 for a more generalized formula it sucks and i hate it

Things that are equivalent. A is invertible $A\vec{x}=\vec{b}$ has one solution for any $\vec{b}$ in order to get ONE solution the PIVOTS must be EQUAL amount of ROWS

if $T(\vec{x})=A\vec{x}$

• is t one to one? (yes i mean theres one solution so if its invertible its one t o one)
• is t onto? (yeah, cos it has a pivot on every row)

Is $T(x,y)=(2x-2t,4x+y)$ one to one or onto? standard_matrix(T) -> $\left[\begin{array}{cc}2& -3\\ 4& 1\end{array}\right]$ its invertible so its onto and one to one

$R^n$ is spanned by n linearly independent vectors

A is invertable and $n\times n$ then its columns span $R^n$

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