QUESTIONS ON SYMMETRIC AND SKEW SYMMETRIC MATRIX

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 = 1

aij  =  i - j

a11 =  1 - 1 

a11  =  0

i = 1, j = 2

aij  =  i - j

a12 =  1 - 2 

a12  =  -1

i = 1, j = 3

aij  =  i - j

a13 =  1 - 3 

a13  =  -2

i = 2, j = 1

aij  =  i - j

a21 =  2 - 1 

a21  =  1

i = 2, j = 2

aij  =  i - j

a22 =  2 - 2  

a22  =  0

i = 2, j = 3

aij  =  i - j

a23 =  2 - 3  

a23  =  -1

i = 3, j = 1

aij  =  i - j

a31 =  3 - 1 

a31  =  2

i = 3, j = 2

aij  =  i - j

a32 =  3 - 2

a32  =  1

i = 3, j = 3

aij  =  i - j

a33 =  3 - 3

a33  =  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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