Question

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

Answer 1

Hint: First of all, consider a symmetric matrix of order $$n$$. Use the properties of transpose of the matrix to get the suitable answer for the given problem.

Complete step-by-step answer:
Let $$A$$ be an invertible symmetric matrix of order $$n$$.
$$\therefore A{A}^{-1}=A^{-1}A=I_n..................................................\left(1\right)$$
Now taking transpose on both sides we have,
$$\Rightarrow {\left(A{A}^{-1}\right)}^{\prime }={\left(A^{-1}A\right)}^{\prime }={\left(I_n\right)}^{\prime }$$
By using the formula $${\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }$$, we have
$$\Rightarrow {\left(A^{-1}\right)}^{\prime }{A}^{\prime }=A^{\prime }{\left(A^{-1}\right)}^{\prime }={\left(I_n\right)}^{\prime }$$
As $$A$$ is a symmetric matrix $$A^{\prime }=A$$ and for the identity matrix $${\left(I_n\right)}^{\prime }=I_n$$
$$\begin{gathered} \Rightarrow {\left(A^{-1}\right)}^{\prime }A=A{\left(A^{-1}\right)}^{\prime }=I_n \\ \therefore {\left(A^{-1}\right)}^{\prime }=A^{\prime }\left[\text{Using }\left(1\right)\right] \end{gathered}$$
As the inverse of the matrix is unique $$A^{-1}$$ is symmetric.
Therefore, the inverse of a symmetric matrix is a symmetric matrix.
Thus, the correct option is A. a symmetric matrix

Note: A symmetric matrix is a square matrix that is equal to its transpose. $$A^{-1}$$ exists and is symmetric if and only if $$A$$ is symmetric. Remember this question as a statement for further references.