Question

If \[{D_1}\] and \[{D_2}\] are two 3 x 3 diagonal matrices, then 1,2,3 are correct.
(A) \[{D_1}{D_2}\] is a diagonal matrix
(B) \[{D_1} + {D_2}\] is a diagonal matrix
(C) \[{D_1}^2 + {D_2}^2\]is a diagonal matrix
(D) 1,2,3 are correct

Answer 1

Hint: Diagonal matrices contain only diagonal elements So, first of all, take two any diagonal matrices then after the check option it is diagonal or not? Now multiply the two diagonal matrices \[{D_1}\] and \[{D_2}\] and check it is diagonal matrices. Now add the two diagonal matrices \[{D_1}\]and\[{D_2}\] and check it is diagonal matrices. Now take the square of two diagonal matrices \[{D_1}\] and \[{D_2}\] and add this square then check it is diagonal matrices.

Complete Step-by-step Solution
Take two any diagonal matrices and test rule that showing in option let us assume:
\[{D_1} = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&0&0 \\
  0&{{a_{22}}}&0 \\
  0&0&{{a_{33}}}
\end{array}} \right)\]
\[{D_2} = \left( {\begin{array}{*{20}{c}}
  {{b_{11}}}&0&0 \\
  0&{{b_{22}}}&0 \\
  0&0&{{b_{33}}}
\end{array}} \right)\]
Now multiply the two diagonal matrices \[{D_1}\] and \[{D_2}\]
Since the matrices have only diagonal values therefore, the product matrix will also have only diagonal entries making it also a diagonal matrix.
\[{D_1}{D_2} = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&0&0 \\
  0&{{a_{22}}}&0 \\
  0&0&{{a_{33}}}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
  {{b_{11}}}&0&0 \\
  0&{{b_{22}}}&0 \\
  0&0&{{b_{33}}}
\end{array}} \right)\]
\[ = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}{b_{11}}}&0&0 \\
  0&{{a_{22}}{b_{22}}}&0 \\
  0&0&{{a_{33}}{b_{33}}}
\end{array}} \right)\]
$\therefore $ Diagonal matrices contain only diagonal elements. So, \[{D_1}{D_2}\] is a diagonal matrix.
Now add the two diagonal matrices \[{D_1}\] and \[{D_2}\]
Adding any two diagonal matrices means adding only the diagonal entries of the two matrices which gives us a diagonal matrix.
\[{D_1} + {D_2} = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}&0&0 \\
  0&{{a_{22}}}&0 \\
  0&0&{{a_{33}}}
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
  {{b_{11}}}&0&0 \\
  0&{{b_{22}}}&0 \\
  0&0&{{b_{33}}}
\end{array}} \right)\]
\[ = \left( {\begin{array}{*{20}{c}}
{{a_{11}} + {b_{11}}}&0&0 \\
0&{{a_{22}} + {b_{22}}}&0 \\
0&0&{{a_{33}} + {b_{33}}}
\end{array}} \right)\]
$\therefore $ Diagonal matrices contain only diagonal elements. So, \[{D_1} + {D_2}\] is a diagonal matrix
Now add the two diagonal matrices \[{D_1}^2\] and \[{D_2}^2\].
First, do the square of each matrix by multiplying two same matrices
\[{D_1}^2 + {D_2}^2 = {D_1} \times {D_1} + {D_2} \times {D_2}\]
\[{D_1}^2 + {D_2}^2 = \left( {\begin{array}{*{20}{c}}
{{a_{11}}}&0&0 \\
0&{{a_{22}}}&0 \\
0&0&{{a_{33}}}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&0&0 \\
0&{{a_{22}}}&0 \\
0&0&{{a_{33}}}
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{{b_{11}}}&0&0 \\
0&{{b_{22}}}&0 \\
0&0&{{b_{33}}}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{{b_{11}}}&0&0 \\
0&{{b_{22}}}&0 \\
0&0&{{b_{33}}}
\end{array}} \right)\]
\[ = \left( {\begin{array}{*{20}{c}}
{{a^2}_{11}}&0&0 \\
0&{{a^2}_{22}}&0 \\
0&0&{{a^2}_{33}}
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
{{b^2}_{11}}&0&0 \\
0&{{b^2}_{22}}&0 \\
0&0&{{b^2}_{33}}
\end{array}} \right)\]
Then add these two matrices
\[ = \left( {\begin{array}{*{20}{c}}
{{a^2}_{11} + {b^2}_{11}}&0&0 \\
0&{{a^2}_{33} + {b^2}_{22}}&0 \\
0&0&{{a^2}_{33} + {b^2}_{33}}
\end{array}} \right)\]
$\therefore $ Diagonal matrices contain only diagonal elements. So, the\[{D_1}^2 + {D_2}^2\]is a diagonal matrix.

$\therefore $ Option (D) is the correct option.

Note:
Students many times make mistakes while multiplying to matrices, they should always keep in mind that while multiplying two matrices we multiply respective elements moving from left to right in rows and from top to bottom in columns. Also, the diagonal entries should be written in a way that they go from top left to right bottom and not from top right to left bottom.