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Question: The inverse of a symmetric matrix (if it exists) is A.asymmetric matrix B.a skew-symmetric matri...

The inverse of a symmetric matrix (if it exists) is
A.asymmetric matrix
B.a skew-symmetric matrix
C.a diagonal matrix
D.none of these

Explanation

Solution

Hint : We need to know about the definition and properties of symmetric matrices. Then we should know how inverse is calculated and their important conditions. If a matrix is symmetric thenAT=A{A^T} = A. We know that inverse of a matrix is possible only when∣A∣≠0\left| A \right| \ne 0.

Complete step-by-step answer :
The given matrix is symmetric.
Let A be a symmetric matrix thenAT=A{A^T} = A …… (1)
We know that I=ITI = {I^T}
Also, II can be written as
I=AA−1I = A{A^{ - 1}}
Substituting the value of II in equation 1

(AA−1)=(AA−1)T,(∵I=AA−1) AA−1=(A−1)TAT,(∵(AB)T=BTAT)  \left( {A{A^{ - 1}}} \right) = {\left( {A{A^{ - 1}}} \right)^T},\left( {\because I = A{A^{ - 1}}} \right) \\\ A{A^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}{A^T},\left( {\because {{\left( {AB} \right)}^T} = {B^T}{A^T}} \right) \\\

It can be further simplified as

A−1A=(A−1)TAT,(∵AA−1=A−1A=I) A−1A=(A−1)TA,(∵A=AT)  {A^{ - 1}}A = {\left( {{A^{ - 1}}} \right)^T}{A^T},\left( {\because A{A^{ - 1}} = {A^{ - 1}}A = I} \right) \\\ {A^{ - 1}}A = {\left( {{A^{ - 1}}} \right)^T}A,\left( {\because A = {A^T}} \right) \\\

Multiply both sides with A−1{A^{ - 1}}

A−1A(A−1)=(A−1)TA(A−1) A−1I=(A−1)TI A−1=(A−1)T  {A^{ - 1}}A\left( {{A^{ - 1}}} \right) = {\left( {{A^{ - 1}}} \right)^T}A\left( {{A^{ - 1}}} \right) \\\ {A^{ - 1}}I = {\left( {{A^{ - 1}}} \right)^T}I \\\ {A^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T} \\\

Thus the inverse of a symmetric matrix is also symmetric.

Note : The properties of matrices are important to know and solve such types of problems. There are some matrix properties which we need to know-AA−1=A−1A=I  &  A=ATA{A^{ - 1}} = {A^{ - 1}}A = I\;\& \;A = {A^T}
Also, try to avoid any calculation mistakes.