# Show that B is the inverse of A. A=begin{bmatrix}1 & -1 -1 & 2 end{bmatrix} , B=begin{bmatrix}2 & 1 1 & 1 end{bmatrix}

Show that B is the inverse of A.
$A=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$
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Nathanael Webber
Step 1
Given the matrices
$A=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$
Show that B is the inverse of A.
Step 2
To show that two matrices are inverse it must satisfy the following condition
AB=BA=I
where I is the identity matrix
$AB=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$
$AB=\left[\begin{array}{cc}\left(1\cdot 2\right)+\left(-1\cdot 1\right)& \left(1\cdot 1\right)+\left(-1\cdot 1\right)\\ \left(-1\cdot 2\right)+\left(2\cdot 1\right)& \left(-1\cdot 1\right)+\left(2\cdot 1\right)\end{array}\right]$
$AB=\left[\begin{array}{cc}2-1& 1-1\\ 1-1& 2-1\end{array}\right]$
$AB=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$
$BA=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$
$BA=\left[\begin{array}{cc}\left(2\cdot 1\right)+\left(1\cdot -1\right)& \left(2\cdot -1\right)+\left(1\cdot 2\right)\\ \left(1\cdot 1\right)+\left(1\cdot -1\right)& \left(1\cdot -1\right)+\left(1\cdot 2\right)\end{array}\right]$
$BA=\left[\begin{array}{cc}2-1& -2+2\\ 1-1& -1+2\end{array}\right]$
$BA=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$
So AB=BA=I
hence B is the inverse of A.
Jeffrey Jordon