# Inverse of a diagonal non-singular matrix is:

$A.$ Symmetric matrix

$B.$ Skew-symmetric matrix

$C.$ Diagonal matrix

$D.$ Scalar matrix

Answer

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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.

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.

Last updated date: 19th Sep 2023

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