## About "Questions on Symmetric and Skew Symmetric Matrix"

Questions on Symmetric and Skew Symmetric Matrix :

Here we are going to see some practice questions on symmetric and skew symmetric matrix.

What is symmetric and skew symmetric matrix ?

A square matrix A is said to be symmetric if AT = A.

A square matrix A is said to be skew-symmetric if AT = −A.

Let us look into some problems to understand the concept.

Question 1 :

Construct the matrix A  =  [aij]3x3, where aij  =  i - j. State whether A is symmetric or skew-symmetric

Solution :

From the given question, we come to know that we have to construct a matrix with 3 rows and 3 columns.

 i = 1, j = 1aij  =  i - ja11 =  1 - 1  a11  =  0 i = 1, j = 2aij  =  i - ja12 =  1 - 2 a12  =  -1 i = 1, j = 3aij  =  i - ja13 =  1 - 3 a13  =  -2 i = 2, j = 1aij  =  i - ja21 =  2 - 1  a21  =  1 i = 2, j = 2aij  =  i - ja22 =  2 - 2  a22  =  0 i = 2, j = 3aij  =  i - ja23 =  2 - 3  a23  =  -1 i = 3, j = 1aij  =  i - ja31 =  3 - 1  a31  =  2 i = 3, j = 2aij  =  i - ja32 =  3 - 2a32  =  1 i = 3, j = 3aij  =  i - ja33 =  3 - 3a33  =  0

So, the matrix A with order 3 x 3 is

Now let us check whether it is symmetric or skew symmetric matrix.

Hence it is skew symmetric matrix.

Question 2 :

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

Solution :

If A and B are symmetric matrices, then

AT  =  A and BT  =  B

From the given question, we have to understand that we have to prove AB  =  BA if AB is symmetric matrix.

If AB is symmetric matrix, then we have to prove AB  =  BA. So, let us prove them as two cases.

Case 1 :

Prove that : AB  =  BA

Given :   AB is symmetric

If AB is symmetric,

then (AB)T  =  AB

By using transpose law,

BTAT  =  AB

(B =  B and AT  =  A)

BA  =  AB

Hence proved.

Case 2 :

Prove that : AB is symmetric

Given :   AB  =  BA

Let us take transpose for AB

(AB)=  BT AT

(AB)T  =  BA

From the given information, AB  =  BA.So let us replace BA as AB.

(AB)T  =  AB

Hence proved.

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