# Find the products AB and BA for the diagonal matrices. A=begin{bmatrix}2 & 0 0 & -3 end{bmatrix} , B=begin{bmatrix}-5 & 0 0 & 4 end{bmatrix}

Find the products AB and BA for the diagonal matrices.
$A=\left[\begin{array}{cc}2& 0\\ 0& -3\end{array}\right],B=\left[\begin{array}{cc}-5& 0\\ 0& 4\end{array}\right]$
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SchulzD
Step 1
Given matrix:
$A=\left[\begin{array}{cc}2& 0\\ 0& -3\end{array}\right],B=\left[\begin{array}{cc}-5& 0\\ 0& 4\end{array}\right]$
Step 2
Now, $AB=\left(\begin{array}{cc}2& 0\\ 0& -3\end{array}\right)\left(\begin{array}{cc}-5& 0\\ 0& 4\end{array}\right)$
Multiply the rows of the first matrix by the columns of the second matrix :
$=\left(\begin{array}{cc}2\cdot \left(-5\right)+0\cdot 0& 2\cdot 0+0\cdot 4\\ 0\cdot \left(-5\right)+\left(-3\right)\cdot 0& 0\cdot 0+\left(-3\right)\cdot 4\end{array}\right)$
$=\left(\begin{array}{cc}-10& 0\\ 0& -12\end{array}\right)$
Step 3
and
$BA=\left(\begin{array}{cc}-5& 0\\ 0& 4\end{array}\right)\left(\begin{array}{cc}2& 0\\ 0& -3\end{array}\right)$
Multiply the rows of the first matrix by the columns of the second matrix : $=\left(\begin{array}{cc}\left(-5\right)2+0\cdot 0& \left(-5\right)\cdot 0+0\cdot \left(-3\right)\\ 0\cdot 2+4\cdot 0& 0\cdot 0+4\cdot \left(-3\right)\end{array}\right)$
$=\left(\begin{array}{cc}-10& 0\\ 0& -12\end{array}\right)$
Jeffrey Jordon