Question

If p=[ x00 0y0 00z ] and q=[a00 0b0 00c] Show that the product of the given diagonal matrices is commutative

Answer 1

Given : $p = \left[\begin{array}{ccc}x&0&0\\0&y&0\\0&0&z\end{array}\right]$   $q = \left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&b\end{array}\right]$

To show :  Product of given Diagonal matrices is Commutative

Solution:

Commutative Property  

p * q  =  q * p

$p = \left[\begin{array}{ccc}x&0&0\\0&y&0\\0&0&z\end{array}\right]$

$q = \left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&b\end{array}\right]$

p * q =  $\left[\begin{array}{ccc}x&0&0\\0&y&0\\0&0&z\end{array}\right]$  $\left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&c\end{array}\right]$

p * q =  $\left[\begin{array}{ccc}ax&0&0\\0&by&0\\0&0&cz\end{array}\right]$

q * p  =  $\left[\begin{array}{ccc}a&0&0\\0&b&0\\0&0&c\end{array}\right]$  $\left[\begin{array}{ccc}x&0&0\\0&y&0\\0&0&z\end{array}\right]$

q * p  = $\left[\begin{array}{ccc}ax&0&0\\0&by&0\\0&0&cz\end{array}\right]$

$\left[\begin{array}{ccc}ax&0&0\\0&by&0\\0&0&cz\end{array}\right]$  = $\left[\begin{array}{ccc}ax&0&0\\0&by&0\\0&0&cz\end{array}\right]$

p * q  = q * p

Hence given Diagonal matrices is  commutative

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