GATE CSE 1994 | Question: 1.2

Question

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

• A. $AA'=I$
• B. $A=A^{-1}$
• C. $AB=BA$
• D. $(AB)'=BA$

Comments

Isn't option C and D both are true?

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Matrix multiplication is not commutative. So only D.

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Option $A$ and $B$ are incorrect  because  It is not given in question that matrix $A$ is invertible, it may be the case that $A$ is singular matrix.

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Option $C$  is incorrect because Matrix multiplicaton are non commutative

Answer 1

A 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!

Comments

Is there any property of A = A' , If  A is a symmetric matrix??

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it can be easily seen from the definition of the symmetric matrix. (Aij = Aji)

 

take an example 

 2    3

 3    4

its transpose will be 

 2    3

 3    4

 

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A.Orthagonal matrix property

B.Involuntry matrix property

D.Symmetric matrix property

Answer 2

Answer: D

Given A = A' and B = B'

(AB)' = B'A' = BA

Answer 3

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