# Find the products AB and BA for the diagonal matrices. A=begin{bmatrix}3 & 0 &0 0 & -5&0 0&0&0 end{bmatrix}, B=begin{bmatrix}-7 & 0 &0 0 &4&0 0&0&12 end{bmatrix}

Tazmin Horton 2021-01-02 Answered
Find the products AB and BA for the diagonal matrices.
$A=\left[\begin{array}{ccc}3& 0& 0\\ 0& -5& 0\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{ccc}-7& 0& 0\\ 0& 4& 0\\ 0& 0& 12\end{array}\right]$
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SchulzD
Step 1
Given the diagonal matrix:
$A=\left[\begin{array}{ccc}3& 0& 0\\ 0& -5& 0\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{ccc}-7& 0& 0\\ 0& 4& 0\\ 0& 0& 12\end{array}\right]$
Step 2
Multiply the given matrix:
$AB=\left[\begin{array}{ccc}3& 0& 0\\ 0& -5& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}-7& 0& 0\\ 0& 4& 0\\ 0& 0& 12\end{array}\right]$
$=\left[\begin{array}{ccc}3\left(-7\right)+0\cdot 0+0\cdot 0& 3\cdot 0+0\cdot 4+0\cdot 0& 3\cdot 0+0\cdot 0+0\cdot 12\\ 0\cdot \left(-7\right)+\left(-5\right)\cdot 0+0\cdot 0& 0\cdot 0+\left(-5\right)\cdot 4+0\cdot 0& 0\cdot 0+\left(-5\right)\cdot 0+0\cdot 12\\ 0\cdot \left(-7\right)+0\cdot 0+0\cdot 0& 0\cdot 0+0\cdot 4+0\cdot 0& 0\cdot 0+0\cdot 0+0\cdot 12\end{array}\right]$

$=\left[\begin{array}{ccc}-21& 0& 0\\ 0& -20& 0\\ 0& 0& 0\end{array}\right]$ Since the multiplication of diagonal matrices is commutative.
That is
If A and B are diagonal matrices, then AB = BA.
Thus,
$AB=BA=\left[\begin{array}{ccc}-21& 0& 0\\ 0& -20& 0\\ 0& 0& 0\end{array}\right]$
Jeffrey Jordon