What is symmetric and skew symmetric matrix ?
For any square matrix A with real number entries, A+ AT is a symmetric matrix and A− AT 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 AT, then we have to multiply it by 1/2
Now we have to subtract the matrices A and AT, 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 AT
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 AT 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
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