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 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 the above

Answer 1

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

$D_1^2+D_2^2$ is a diagonal matrix


Let $D_1=\text{diag}(a_1,a_2,a_3)$ and
$D_2=\text{diag}(b_1,b_2,b_3)$

Now, $D_1{D}_2=\left[\begin{array}{ccc}{a}_1& 0& 0\\ 0& a_2& 0\\ 0& 0& a_3\end{array}\right]\left[\begin{array}{ccc}{b}_1& 0& 0\\ 0& b_2& 0\\ 0& 0& b_3\end{array}\right]$

$\Rightarrow D_1{D}_2=\left[\begin{array}{ccc}{a}_1{b}_1& 0& 0\\ 0& a_2{b}_2& 0\\ 0& 0& a_3{b}_3\end{array}\right]$

$\therefore D_1{D}_2=\text{diag}(a_1{b}_1,a_2{b}_2,a_3{b}_3)$
which is a diagonal matrix.

$\Rightarrow D_1{D}_2=\text{diag}(b_1{a}_1,b_2{a}_2,b_3{a}_3)$
$=\left[\text{diag}\right(b_1,b_2,b_3\left)\right]\left[\text{diag}\right(a_1,a_2,a_3\left)\right]$
$=D_2{D}_1$
$\therefore D_1{D}_2=D_2{D}_1$

$(D_1{)}^2=(\text{diag}(a_1,a_2,a_3){)}^2$
$=\text{diag}(a_1^2,a_2^2,a_3^2)$

$(D_2{)}^2=\text{diag}(b_1,b_2,b_3){)}^2$
$=\text{diag}(b_1^2,b_2^2,b_3^2)$

$D_1^2+D_2^2=\text{diag}(a_1^2+b_1^2,a_2^2+b_2^2,a_3^2+b_3^2)$
which is also a diagonal matrix.