First slide

Applications of determinant

Question

The inverse of a symmetric matrix (if it exists) is

Moderate

A

a symmetric matrix
B

a skew-symmetric matrix
C

a diagonal matrix
D

none of these

Solution

Let A be an invertible symmetric matrix.
We have $A A^{- 1} = A^{- 1} A = I_{n}$
$\begin{aligned} \Rightarrow {(A A^{- 1})}^{'} = {(A^{- 1} A)}^{'} = {(I_{n})}^{'} \\ \Rightarrow {(A^{- 1})}^{'} A^{'} = A^{'} {(A^{- 1})}^{'} = I_{n} \\ \Rightarrow {(A^{- 1})}^{'} A = A {(A^{- 1})}^{'} = I_{n} \end{aligned}$
${(A^{- 1})}^{'} = A^{- 1}$ [inverse of a matrix is unique]

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