Question

Inverse of a diagonal non-singular matrix is:
$A.$ Symmetric matrix
$B.$ Skew-symmetric matrix
$C.$ Diagonal matrix
$D.$ Scalar matrix

Answer 1

Hint: - Just consider the cases given by taking an example in mind to solve such problems. These questions don’t need lots of working.

Taking an example of a diagonal matrix and finding its inverse we check the following result.
A = \left( {\begin{array}{*{20}{c}} 2&0&0 \\\ 0&3&0 \\\ 0&0&4 \end{array}} \right) Where $A$is a diagonal matrix.
{A^{ - 1}} = \left( {\begin{array}{*{20}{c}} {\dfrac{1}{2}}&0&0 \\\ 0&{\dfrac{1}{3}}&0 \\\ 0&0&{\dfrac{1}{4}} \end{array}} \right) and${A^{ - 1}}$ is the inverse of a diagonal matrix.
We find by an example that the inverse of a diagonal matrix is also a diagonal matrix.
Inverse of a nonsingular diagonal matrix is a nonsingular diagonal matrix with all the diagonal elements inverted. Therefore, the resultant invertible matrix is a diagonal matrix.
So the correct option is C.

Note: In linear algebra, a diagonal matrix has values of entries outside the main diagonal as zero; the term usually refers to a square matrix. In the above question it is easier to check the results by example rather than going by finding formulae.