Q. 10 A3.9( 37 Votes )

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:


Answer :

As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”



Now, on adding A and A’ we will get,



.



.


.


.


on subtracting A’ from A we will get,











Now, Add M and N, we get,


.





us, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.


Ans. Hence proved


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