(A) Given that $A$ is a symmetric matrix, we have $A^T = A$. If the inverse $A^{-1}$ exists, we use the property $(A^{-1})^T = (A^T)^{-1}$. Substituti

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$A^{-1}$ is symmetric, if it exists

(A) Given that $A$ is a symmetric matrix, we have $A^T = A$. If the inverse $A^{-1}$ exists, we use the property $(A^{-1})^T = (A^T)^{-1}$. Substituting $A^T = A$ into the equation, we get $(A^{-1})^T = (A)^{-1} = A^{-1}$. Since the transpose of $A^{-1}$ is equal to $A^{-1}$, it follows that $A^{-1}$ is a symmetric matrix, provided it exists.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Adjoint and inverse of matrices

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If $A$ is a symmetric matrix with real entries, then

TS EAMCET 2023 All TS EAMCET

A
$A^{-1}$ is symmetric, if it exists
B
$A^{-1}$ always exists and is symmetric
C
$A^{-1}$ is skew-symmetric, if it exists
D
$A^{-1}$ always exists and is skew-symmetric

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3 and 4 .Determinants and Matrices

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Adjoint and inverse of matrices

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If $A$ is a symmetric matrix with real entries, then

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