Example for Skew Symmetric Matrix :
Here we are going to see some example problems on 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 :
If A
is a matrix such that AA^{T} = 9I , find the values of x and y.
Solution :
From the given matrix A, first let us find the transpose of matrix A^{T}.
Here I stands for identity matrix with order 3 x 3.
By equating the corresponding terms, we get
x + 2y + 4 = 0 ------(1)
2x - 2y + 2 = 0 ------(2)
By adding the first and second equation, we may eliminate Y.
(x + 2y + 4) + (2x - 2y + 2) = 0 + 0
x + 2x + 4 + 2 = 0
3x + 6 = 0
3x = -6
x = -6/3 = -2
So, the value of x is -2.
Now we have to apply the value of x in the first equation, we may get y.
-2 + 2y + 4 = 0
2 + 2y = 0
2y = -2
y = -1
Hence the values of x and y are -2 and -1 respectively.
Question 2 :
For what value of x, the matrix
is skew-symmetric
Solution :
A square matrix A is said to be skew-symmetric if A^{T} = −A.
By equating the corresponding terms, we get the value of x.
-3 = -x^{3}
x^{3 } = 3
x = 3^{1/3}
Hence the value of x is 3^{1/3}.
Question 3 :
If A =
is skew-symmetric, find the values of p, q, and r.
By equating the corresponding values, we may find the values of p, q and r respectively.
2 = -p p = -2 |
q^{2} = -q^{2} 2q^{2} = 0 q = 0 |
3 = -r r = -3 |
Hence the values of p, q and r are -2, 0 and -3 respectively.
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