How to Express the Given Matrix as Sum of Symmetric and Skew Symmetric :
Here we are going to see how to express the given matrix as the sum of symmetric and skew symmetric matrix.
What is symmetric and skew symmetric matrix ?
For any square matrix A with real number entries, A+ A^{T} is a symmetric matrix and A− A^{T} is a skew-symmetric matrix.
Any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
Let A be a square matrix. Then, we can write
Let us look into some problems to understand the concept.
Question 1 :
Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
Solution :
First let us add the matrices A and A^{T}, then we have to multiply it by 1/2
Now we have to subtract the matrices A and A^{T}, then we have to multiply it by 1/2
By adding the above two matrices, we get the original question.
Hence proved.
(ii) From the given matrix A, we have to find A^{T}
So far we get symmetric matrix. Now we are going to find skew symmetric matrix.
By adding (1) and (2), we get
Hence proved.
Question 3 :
Find the matrix A such that
Solution :
Let the required matrix A^{T} be
By equating the corresponding terms, we get
2a-d = -1 ----(1)
2b - e = -8 ----(2)
2c - f = -10 ----(3)
a = 1, b = 2 and c = -5
By applying the value of a in the first equation, we get the value d.
2(1) - d = -1
2 - d = -1
-d = -1 - 2
-d = -3
d = 3
By applying the value of b in the second equation, we get the value e.
2(2) - e = -8
4 - e = -8
-e = -8 - 4
-e = -12
e = 12
By applying the value of c in the third equation, we get the value f.
2(-5) - f = -10
-10 - f = -10
-f = -10 + 10
f = 0
After having gone through the stuff given above, we hope that the students would have understood "How to Express the Given Matrix as Sum of Symmetric and Skew Symmetric".
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