Question

If Di and D2 are two 3 x 3 diagonal matrices, then
dia
(A) D1,D2 is a diagonal matrix
(B) D1D2 = D2D1
(C) D^2 1+ D^2 2is a diagonal matrix
(D) None of these

Answer 1

Given :  D1 and D2 are two 3 x 3 diagonal matrices

To find :  Which all are true  

Solution:

Diagonal Matrix   has all elements = 0 other than diagonals

D1,D2 is a diagonal matrix

Let say

D1  =  $\left[\begin{array}{ccc}1&0&0\\0&2&0\\0&0&3\end{array}\right]$

& D2  =  $\left[\begin{array}{ccc}4&0&0\\0&5&0\\0&0&6\end{array}\right]$

D1D2  =  $\left[\begin{array}{ccc}4&0&0\\0&10&0\\0&0&18\end{array}\right]$  

D1D2 - Diagonal Matrix

D2D1 =  $\left[\begin{array}{ccc}4&0&0\\0&10&0\\0&0&18\end{array}\right]$

D1D2 = D2D1  

D1² =   $\left[\begin{array}{ccc}1&0&0\\0&4&0\\0&0&9\end{array}\right]$

D2²  =  $\left[\begin{array}{ccc}16&0&0\\0&25&0\\0&0&36\end{array}\right]$

D1²  + D2² =  $\left[\begin{array}{ccc}17&0&0\\0&29&0\\0&0&45\end{array}\right]$  

D1²  + D2²  is Diagonal Matrix

Hence A , B & C are all true :

None of these is False

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