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26 votes
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Let $A$ and $B$ be real symmetric matrices of size $n \times n$. Then which one of the following is true?

  1. $AA'=I$
  2. $A=A^{-1}$
  3. $AB=BA$
  4. $(AB)'=BA$
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3 Answers

Best answer
28 votes
28 votes

symmetric matrix is a square matrix that is equal to its transpose.

$(AB)' = B'A' =BA$ as both $A$ and $B$ are symmetric matrices, hence $B' = B$ and $A'=A$

So, (D) is correct option!

Why is $(C)$ not correct option? see the following example:

$\begin{bmatrix}1&1\\1&1\end{bmatrix}\begin{bmatrix}1&0\\0&2\end{bmatrix}=\begin{bmatrix}1&2\\1&2\end{bmatrix}$

$\begin{bmatrix}1&0\\0&2\end{bmatrix}\begin{bmatrix}1&1\\1&1\end{bmatrix}=\begin{bmatrix}1&1\\2&2\end{bmatrix}$

There are two symmetric matrices given of size $2\times2$ and $AB != BA$. Therefore (C) is not a correct option!

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