Question

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

Answer 1

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 then${A^T} = A$. We know that inverse of a matrix is possible only when$\left| A \right| \ne 0$.

Complete step-by-step answer:
The given matrix is symmetric.
Let A be a symmetric matrix then${A^T} = A$ …… (1)
We know that \[I = {I^T}\]
Also, $I$ can be written as
\[I = A{A^{ - 1}}\]
Substituting the value of $I$ in equation 1
\[
  \left( {A{A^{ - 1}}} \right) = {\left( {A{A^{ - 1}}} \right)^T},\left( {\because I = A{A^{ - 1}}} \right) \\
  A{A^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}{A^T},\left( {\because {{\left( {AB} \right)}^T} = {B^T}{A^T}} \right) \\
 \]
It can be further simplified as
\[
  {A^{ - 1}}A = {\left( {{A^{ - 1}}} \right)^T}{A^T},\left( {\because A{A^{ - 1}} = {A^{ - 1}}A = I} \right) \\
  {A^{ - 1}}A = {\left( {{A^{ - 1}}} \right)^T}A,\left( {\because A = {A^T}} \right) \\
 \]
Multiply both sides with \[{A^{ - 1}}\]
\[
  {A^{ - 1}}A\left( {{A^{ - 1}}} \right) = {\left( {{A^{ - 1}}} \right)^T}A\left( {{A^{ - 1}}} \right) \\
  {A^{ - 1}}I = {\left( {{A^{ - 1}}} \right)^T}I \\
  {A^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T} \\
 \]
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-\[A{A^{ - 1}} = {A^{ - 1}}A = I\;\& \;A = {A^T}\]
Also, try to avoid any calculation mistakes.