Question

If A is an invertible symmetric matrix,then ${\text{A}}^{\text{- 1}}$ is

• A. Symmetric
• B. Skew symmetric
• C. A diagonal
• D. none of these

Answer 1

The correct option is A Symmetric

Let A be a invertible matrix and symmetric matrix.

$\therefore A{A}^{-1}=I$

$\Rightarrow \left(A{A}^{-1}\right)=(A{A}^{-1}{)}^T$

$\because (AB{)}^T=B^T{A}^T$

$\Rightarrow A{A}^{-1}=(A^{-1}{)}^T{A}^T$ and $A{A}^{-1}=A^{-1}A=I$

$\Rightarrow A^{-1}A=(A^{-1}{)}^{T}A$ and $A=A^T$

multiplying by $A^{-1}$ on both sides

$\Rightarrow A^{-1}A{A}^{-1}=(A^{-1}{)}^{T}A{A}^{-1}$

$\Rightarrow A^{-1}I=(A^{-1}{)}^{T}I$

$\Rightarrow A^{-1}=(A^{-1}{)}^T$

$\therefore A^{-1}$ is a symmetric matrix.