 # To illustrate the multiplication of matrices, and also the fact that matrix multiplication is not necessarily commutative, consider the matrices A=begin{bmatrix}1 & -2&1 0 & 2&-12&1&1 end{bmatrix} B=begin{bmatrix}2 & 1&-1 1 & -1&02&-1&1 end{bmatrix} amanf 2020-11-30 Answered
To illustrate the multiplication of matrices, and also the fact that matrix multiplication is not necessarily commutative, consider the matrices
$A=\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& -1\\ 2& 1& 1\end{array}\right]$
$B=\left[\begin{array}{ccc}2& 1& -1\\ 1& -1& 0\\ 2& -1& 1\end{array}\right]$
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Step 1
$A=\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& -1\\ 2& 1& 1\end{array}\right]$
$B=\left[\begin{array}{ccc}2& 1& -1\\ 1& -1& 0\\ 2& -1& 1\end{array}\right]$
$AB=\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& -1\\ 2& 1& 1\end{array}\right]\left[\begin{array}{ccc}2& 1& -1\\ 1& -1& 0\\ 2& -1& 1\end{array}\right]$
$=\left[\begin{array}{ccc}2-2+2& 1+2-1& -1+1\\ 2-2& -2+1& -1\\ 4+1+2& 2-1-1& -2+1\end{array}\right]$
$=\left[\begin{array}{ccc}2& 2& 0\\ 0& -1& -1\\ 7& 0& -1\end{array}\right]$
Step 2
Acc. to commutative property:
AB=BA
$BA=\left[\begin{array}{ccc}2& 1& -1\\ 1& -1& 0\\ 2& -1& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& -1\\ 2& 1& 1\end{array}\right]$
$=\left[\begin{array}{ccc}0& -3& 0\\ 0& -4& 2\\ 4& -5& 4\end{array}\right]$ since here $BA\ne AB$
So , matrix is not necessarily commutative
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