Question

The inverse of a diagonal matrix is a:
A) Symmetric matrix
B) Skew-symmetric matrix
C) Diagonal matrix
D) None of the above

Answer 1

Hint:
Here, we have to find the matrix which is the inverse of the diagonal matrix. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A square matrix in which every element except the main diagonal elements is zero is called a Diagonal Matrix.

Formula Used:
The inverse of a diagonal matrix is given by \[{D^{ - 1}} = \dfrac{1}{{\left| D \right|}}adjD\]

Complete step by step solution:
Let \[D = \left[ {\begin{array}{*{20}{l}}{{a_{11}}}&0&0\\0&{{a_{22}}}&0\\0&0&{{a_{33}}}\end{array}} \right]\] .
Now, we will find the determinants of the diagonal matrix. So,
\[\left| D \right| = {a_{11}}\left[ {\begin{array}{*{20}{l}}{{a_{22}}}&0\\0&{{a_{33}}}\end{array}} \right] + 0\left[ {\begin{array}{*{20}{l}}0&0\\0&{{a_{33}}}\end{array}} \right] + 0\left[ {\begin{array}{*{20}{l}}0&{{a_{22}}}\\0&0\end{array}} \right]\]
\[ \Rightarrow \left| D \right| = {a_{11}}{a_{22}}{a_{33}}\]
Now we will find the adjugate matrix. So,
\[adjD = \left[ {\begin{array}{*{20}{l}}{{a_{22}}{a_{33}}}&0&0\\0&{{a_{11}}{a_{33}}}&0\\0&0&{{a_{11}}{a_{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{\dfrac{1}{{{a_{11}}}}}&0&0\\0&{\dfrac{1}{{{a_{22}}}}}&0\\0&0&{\dfrac{1}{{{a_{33}}}}}\end{array}} \right]\]
The inverse of a diagonal matrix is given by \[{D^{ - 1}} = \dfrac{1}{{\left| D \right|}}adjD\].
Substituting the values of \[\left| D \right|\] and \[adjD\] in the formula \[{D^{ - 1}} = \dfrac{1}{{\left| D \right|}}adjD\], we get
\[ \Rightarrow {D^{ - 1}} = \dfrac{1}{{{a_{11}}{a_{22}}{a_{33}}}}\left[ {\begin{array}{*{20}{l}}{{a_{22}}{a_{33}}}&0&0\\0&{{a_{11}}{a_{33}}}&0\\0&0&{{a_{11}}{a_{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{l}}{\dfrac{1}{{{a_{11}}}}}&0&0\\0&{\dfrac{1}{{{a_{22}}}}}&0\\0&0&{\dfrac{1}{{{a_{33}}}}}\end{array}} \right]\]
A diagonal matrix has elements only in its diagonal. So the inverse is also having all the non zero elements in the diagonal. So, it will be symmetric and will also be a diagonal matrix.
If the elements on the main diagonal are the inverse of the corresponding element on the main diagonal of the D, then D is a diagonal matrix. If the transpose of a matrix is equal to itself, that matrix is said to be symmetric.
Therefore, the inverse of a diagonal matrix is a Symmetric and Diagonal matrix.

Hence, the correct options are option A and C.

Note:
We should also know the properties of Diagonal matrix to know the inverse of a diagonal matrix. The determinant of diagonal \[\left( {{a_1},{\rm{ }}...,{\rm{ }}{a_n}} \right)\]is the product \[{a_1} \cdot ...... \cdot {a_n}.\] The adjugate of a diagonal matrix is again diagonal. A square matrix is diagonal if and only if it is triangular and normal. Any square diagonal matrix is also a symmetric matrix. A symmetric diagonal matrix can be defined as a matrix that is both upper- and lower-triangular. The identity matrix \[{I_n}\] and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal. So, the inverse of the diagonal matrix is a symmetric matrix and diagonal matrix.