Question

If D${}_1$ and D${}_2$ are two 3 $\times$ 3 diagonal matrices, then which of the following is/are true?

• A. $D_1{D}_2$ is a diagonal matrix
• B. $D_1{D}_2=D_2{D}_1$
• C. $D_1^2+D_2^2$ is a diagonal matrix
• D. None of these

Answer 1

Answer: (A), (B), (C)

$D_1{D}_2$ is a diagonal matrix
$D_1{D}_2=D_2{D}_1$
$D_1^2+D_2^2$ is a diagonal matrix

Let $D_1=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$ and $D_2=\left[\begin{array}{ccc}d& 0& 0\\ 0& e& 0\\ 0& 0& f\end{array}\right]$

$D_1{D}_2=\left[\begin{array}{ccc}ad& 0& 0\\ 0& be& 0\\ 0& 0& cf\end{array}\right]$ is also a diagonal matrix

$D_2{D}_1=\left[\begin{array}{ccc}da& 0& 0\\ 0& eb& 0\\ 0& 0& fc\end{array}\right]$

Hence $D_1{D}_2=D_2{D}_1$

$D_1^2=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& b^2& 0\\ 0& 0& c^2\end{array}\right]$

$D_2^2=\left[\begin{array}{ccc}{d}^2& 0& 0\\ 0& e^2& 0\\ 0& 0& f^2\end{array}\right]$

$D_1^2+D_2^2=\left[\begin{array}{ccc}{a}^2+d^2& 0& 0\\ 0& b^2+e^2& 0\\ 0& 0& c^2+f^2\end{array}\right]$

Hence $D_1^2+D_2^2$ is also a diagonal matrix.