lwfrgin

2021-01-31

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A=\left[\begin{array}{cc}-4& 0\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}-2& 4\\ 0& 1\end{array}\right]$

Yusuf Keller

Given
$A=\left[\begin{array}{cc}-4& 0\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}-2& 4\\ 0& 1\end{array}\right]$
Now,
$AB=\left[\begin{array}{cc}-4& 0\\ 1& 3\end{array}\right]\left[\begin{array}{cc}-2& 4\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}-4\left(-2\right)+0\left(0\right)& -4\left(4\right)+0\left(1\right)\\ 1\left(-2\right)+3\left(0\right)& 1\left(4\right)+3\left(1\right)\end{array}\right]$
$=\left[\begin{array}{cc}8& -16\\ -2& 7\end{array}\right]$
Step 2
$BA=\left[\begin{array}{cc}-2& 4\\ 0& 1\end{array}\right]\left[\begin{array}{cc}-4& 0\\ 1& 3\end{array}\right]$
$=\left[\begin{array}{cc}-2\left(-4\right)+4\left(1\right)& -2\left(0\right)+4\left(3\right)\\ 0\left(-4\right)+1\left(1\right)& 0\left(0\right)+1\left(3\right)\end{array}\right]$
$=\left[\begin{array}{cc}12& 12\\ 1& 3\end{array}\right]$
Clearly the products AB and BA are not identity matrices, therefore B is not the multiplicative inverse of A.

Jeffrey Jordon