Question

The inverse of a symmetric matrix is
A.Symmetric
B.Skew-symmetric
C.Diagonal
D.None of these

Answer 1

Hint Here, we will start by assuming that \[A\] be a symmetric matrix, then \[{A^T} = A\]. Then we will use the property of inverse and transpose of a matrix, \[{\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}\]after taking inverse to find the required value.

Complete step-by-step answer:
We are given that the matrix is a symmetric matrix.
Let us assume that \[A\] be a symmetric matrix, then \[{A^T} = A\].
Taking the inverse of the above equation on both sides, we get
\[ \Rightarrow {\left( {{A^T}} \right)^{ - 1}} = {A^{ - 1}}\]
Using the property of inverse and transpose of a matrix, \[{\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}\]in the above equation, we get
\[ \Rightarrow {\left( {{A^{ - 1}}} \right)^T} = {A^{ - 1}}\]
Thus, \[{A^{ - 1}}\] is a symmetric matrix.
We know that the inverse of a symmetric matrix is a symmetric matrix, if it exists.
Hence, option A is correct.

Note In solving these types of questions, the key concept of solving is we should have knowledge of inverses of symmetric matrix and meaning of symmetric matrix. There are many methods to find the inverse of a matrix, which depends on the type of questions. But since this question is a general as having knowledge about inverses is enough.