1. Symmetric and Skew Symmetric
Matrices
Theorem and Proof

2. Symmetric and Skew – Symmetric Matrix
A square matrix A is called a symmetric matrix, if AT = A.
A square matrix A is called a skew- symmetric matrix, if AT = - A.
Any square matrix can be expressed as the sum of a symmetric and a skew- symmetric matrix.

3. Theorem 1 For any square matrix A with real number entries,
A + A ′ is a symmetric matrix and
A – A ′ is a skew symmetric matrix.

4. Proof Let B = A + A ′, then B′ =(A + A′)′
=A′ + (A′ )′ (as (A + B) ′ = A ′ + B ′ )
=A′ + A (as (A ′) ′ = A)
=A + A′ (as A + B = B + A) =B
Therefore B = A + A′ is a symmetric matrix
Now let C = A – A′
C′ = (A – A′ )′ = A ′ – (A′)′ (Why?)
=A′ – A (Why?)
= – (A – A ′) = – C
Therefore C = A – A′ is a skew symmetric matrix.

5. Theorem 2
Any square matrix can be expressed as the sum of a symmetric and a
skew symmetric matrix.

6. Proof Let A be a square matrix, then we can writen as
From the Theorem 1, we know that (A + A ′ ) is a symmetric matrix and (A – A ′) is a skew symmetric matrix. Since for any matrix A, ( kA)′ = kA′, it follows that
is symmetric matrix and
is skew symmetric matrix.
Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

7. Example Show that A= is a skew-symmetric matrix.
Solution :
As AT = - A, A is a skew – symmetric matrix

8. Example Express the matrix as the sum of a
symmetric and a skew- symmetric matrix.
Solution :

9. Solution Cont.

10. Solution Cont.
Therefore, P is symmetric and Q is skew- symmetric . Further, P+Q = A
Hence, A can be expressed as the sum of a symmetric and a skew -symmetric matrix.

11. ASSESSMENT
(Symmetric and Skew Symmetric Matrices)

12. Express the matrix as the sum of a
Question 1:
Express the matrix as the sum of a
symmetric and askew symmetric matrix.
Question 2: Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(II)
(I)
(III)

13. For the matrix , verify that (i) (A + A′) is a symmetric matrix
Question 3:
For the matrix , verify that
(i) (A + A′) is a symmetric matrix
(ii) (A – A ′) is a skew symmetric matrix
Question 4: