- Books Name
- Mathmatics Book Based on NCERT
- Publication
- KRISHNA PUBLICATIONS
- Course
- CBSE Class 12
- Subject
- Mathmatics
Symmetric and Skew Symmetric Matrices
A square matrix A = [aij] is said to be symmetric if the transpose of A is equal to A, that is, [aij] = [aji] for all possible values of i and j.
A square matrix A = [aij] is a skew-symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Also, if we substitute i = j, we have aii = – aii and thus, 2aii = 0 or aii = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.
Properties of Symmetric Matrix
- Addition and difference of two symmetric matrices results in symmetric matrix.
- If A and B are two symmetric matrices and they follow the commutative property,
i.e. AB =BA, then the product of A and B is symmetric.
- If matrix A is symmetric then An is also symmetric, where n is an integer.
- If A is a symmetrix matrix then A-1 is also symmetric.
Properties of Skew Symmetric Matrix
- When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
- Scalar product of skew-symmetric matrix is also a skew-symmetric matrix.
- The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
- When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible.
- The determinant of skew symmetric matrix is non-negative
Theorem : How do you prove symmetric and skew symmetric matrix?
Example
Properties of Symmetric and Skew-Symmetric Matrices
There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,
1. If matrix A is a square matrix then (A + At) is always symmetric.
Prove:
To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix
So, let’s find the transpose of (A + At)
= (A + At)t
= At + (At)t
= At + A [here, (At)t = A]
= (A + At)
So, this is the same as the given matrix, so it is symmetric.
2. If matrix A is a square matrix then (A – At) is always skew-symmetric.
Prove:
To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix
So, let’s find the transpose of (A – At)
= (A − At)t
= At − (At)t
= At − A [here, (At)t = A]
= − (A − At)
So, this form is the negative of the given matrix, so it is skew-symmetric.
Theorem 1:
Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.
Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.
Proof: Let A = [aij]mxm
We know that A + A' : Symmetric matrix
A - A' : Skew Symmetric matrix
= ½(A + A' ) + ½ (A - A')
= (½A + ½A') + (½A - ½A') Using kA = k(A)
= ½A + ½A = A.
Theorem 3: Inverse of matrix A is unique.
Proof:
Let B and C be two inverses of A.
i.e., AxB=BxA=I
and AxC =CxA=1
Now, B=BI = B (AxC) = ( BxA) X C = I xC = C
So, inverses are unique.