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 A^{T} = A.
A square matrix A is said to be skew-symmetric if A^{T} = −A.
Let us look into some problems to understand the concept.
Question 1 :
Construct the matrix A = [a_{ij}]_{3x3}, where a_{ij } = 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 a_{ij } = i - j a_{11} = 1 - 1 a_{11 } = 0 |
i = 1, j = 2 a_{ij } = i - j a_{12} = 1 - 2 a_{12 } = -1 |
i = 1, j = 3 a_{ij } = i - j a_{13} = 1 - 3 a_{13 } = -2 |
i = 2, j = 1 a_{ij } = i - j a_{21} = 2 - 1 a_{21 } = 1 |
i = 2, j = 2 a_{ij } = i - j a_{22} = 2 - 2 a_{22 } = 0 |
i = 2, j = 3 a_{ij } = i - j a_{23} = 2 - 3 a_{23 } = -1 |
i = 3, j = 1 a_{ij } = i - j a_{31} = 3 - 1 a_{31 } = 2 |
i = 3, j = 2 a_{ij } = i - j a_{32} = 3 - 2 a_{32 } = 1 |
i = 3, j = 3 a_{ij } = i - j a_{33} = 3 - 3 a_{33 } = 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
A^{T} = A and B^{T} = 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,
B^{T}A^{T} = AB
(B^{T } = B and A^{T} = A)
BA = AB
Hence proved.
Case 2 :
Prove that : AB is symmetric
Given : AB = BA
Let us take transpose for AB
(AB)^{T }= B^{T} A^{T}
(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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