# Symmetric and Skew-Symmetric Matrices

As we are all aware by now that equal matrices have equal dimensions, hence only the square matrices can be symmetric or skew-symmetric form. Now, what is a symmetric matrix and a skew-symmetric matrix? Let’s learn about them in further detail below.

## Symmetric Matrix

A square matrix A is said to be symmetric ifÂ aijÂ =Â ajiÂ for all i and j, whereÂ aij is an element present atÂ (i,j)th position (ith row and jthÂ column in matrix A) and aji is an element present atÂ (j,i)th position (jth row and ithÂ  column in matrix A). In other words, we can say that matrix A is said to be symmetric if the transposeÂ of matrix A is equal to matrix A itself (AT=A). Let’s take an example of a matrix,

It is symmetric matrix becauseÂ aijÂ =Â ajiÂ for all i and j.Â Here, a12Â =Â a21= 3,Â a13Â =Â a31= 8Â andÂ a23Â =Â a32Â = -4Â In other words, the transpose of Matrix A is equal to Matrix AÂ itself (AT=A) which means matrix A is symmetric.

## Â Skew-Symmetric Matrix

Square matrix A is said to be skew-symmetric ifÂ aijÂ =âˆ’aji for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A i.e (ATÂ =âˆ’A). Note that all the main diagonal elements in the skew-symmetric matrix are zero. Let’s take an example of a matrix

It is skew-symmetric matrix becauseÂ aijÂ =âˆ’aji for all i and j. Here, a12 = -6 andÂ a21= -6 which means Â a12=Â âˆ’a21. Similarly, this condition holds true for all other values of i and j.

## Theorem 1

For any square matrix A with real number entries, A + Aâ€² is a symmetric matrix and A â€“ Aâ€² is a skew-symmetric matrix.

Proof: Â Let B =A+Aâ€², then Bâ€²= (A+Aâ€²)â€²

= Â Aâ€² + (Aâ€²)â€² (as (A + B)â€² = Aâ€² + Bâ€²)
= Â Aâ€² +A (as (Aâ€²)â€² =A)
= Â A + Aâ€² (as A + B = B + A) =B
Therefore, B = A+Aâ€²is a symmetric matrix

Now let C = A â€“ Aâ€²
C’ = (Aâ€“Aâ€²)â€²=Aâ€²â€“(Aâ€²)â€² (Why?)
= Aâ€² â€“ A (Why?)
=â€“ (A â€“ Aâ€²) = â€“ C
Hence, A â€“ Aâ€² 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 be a square matrix then, we can write A = 1/2 (A + Aâ€²) + 1/2 (A âˆ’ Aâ€²). From the Theorem 1, we know that (A + Aâ€²) is a symmetric matrix and (A â€“ Aâ€²) is a skew-symmetric matrix.

Since for any matrix A, (kA)â€² = kAâ€², it follows that 1 / 2 (A+Aâ€²) is a symmetric matrix and 1 / 2 (A âˆ’ Aâ€²) is a skew-symmetric matrix. Thus, any squareÂ matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

## Solved Examples for You

Question 1: If A and B are symmetric matrices, then ABA is

1. Symmetric
2. Skew – Symmetric
3. Diagonal
4. Triangular

AnswerÂ : Given A and B are Symmetric Matrices
â‡’ AT = A and BT = B
Now, take (ABA)T
(ABA)T = ATBTAT
(ABA)T = ABA
Hence, ABA is also Symmetric

Question 2: Say true or false:Â If A & B are symmetric matrices of same order then ABÂ âˆ’Â BAÂ is symmetric.

1. True
2. False

Answer : Given, A and B are symmetric matrices, therefore we have:
A’ = A and B’ = B……….(i)
Consider, (AB – BA)’ = (AB)’ – (BA)’……………[ Since, (A – B)’ = A’ – B’]
= B’A’ – A’ B’ ……………[ Since, (AB)’ = B’ A’]
= BA – AB …..[by (i)]
= – (AB – BA)
Therefore,Â (AB – BA)’ = –Â (AB – BA)
Thus,Â (AB – BA) is a skew-symmetric matrix

Question 3:Â  Explain a symmetric matrix?

Answer: Symmetric matrix refers to a matrix in which the transpose is equal to itself.

Question 4: Explain a skew symmetric matrix?

Answer: A matrix can be skew symmetric only if it happens to be square. In case the transpose of a matrix happens to be equal to the negative of itself, then one can say that the matrix is skew symmetric. Therefore, for a matrix to be skew symmetric, A’=-A.

Question 5: What is meant by the inverse of a symmetric matrix?

Answer: The inverse of a symmetric matrix happens to be the same as the inverse of any matrix. As such, any matrix, whose multiplication takes place (from the right or the left) with the matrix in question, results in the production of the identity matrix. An important point to understand is that not all symmetric matrices are invertible.

Question 6: Can we say that a positive definite matrix is symmetric?

Answer: A positive definite matrix happens to be a symmetric matrix that has all positive eigenvalues. Note that all the eigenvalues are real because it’s a symmetric matrix all the eigenvalues are real. Consequently,Â  it makes sense to discuss them being positive or negative.

Share with friends

## Customize your course in 30 seconds

##### Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
Ashhar Firdausi
IIT Roorkee
Biology
Dr. Nazma Shaik
VTU
Chemistry
Gaurav Tiwari
APJAKTU
Physics
Get Started

## Browse

### One response to “Types of Matrices”

1. KAINAT says:

MATHEMATICS WAS TOO DIFFICULT FOR ME BUT WHEN I LEARN FROM TOPPR I FEEL MATHEMATICS IS TOO EASY I LIKE IT