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.

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## Symmetric Matrix

A square matrix A is said to be symmetric if a_{ij} = a_{j}_{i} for all i and j, where a_{ij} is an element present at (i,j)^{th} position (i^{th} row and j^{t}^{h} column in matrix A) and a_{ji} is an element present at (j,i)^{th} position (j^{th} row and i^{th} column in matrix A). In other words, we can say that matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself (A^{T}=A). Let’s take an example of a matrix,

It is symmetric matrix because a_{ij} = a_{j}_{i} for all i and j. Here, a_{12 }= a_{21}= 3, a_{13} = a_{31}= 8 and a_{23 }= a_{32} = -4 In other words, the transpose of Matrix A is equal to Matrix A itself (A^{T}=A) which means matrix A is symmetric.

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## Skew-Symmetric Matrix

Square matrix A is said to be skew-symmetric if a_{ij} =−a_{ji} 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 (A^{T} =−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 because a_{i}_{j} =−a_{ji} for all i and j. Here, a_{12} = -6 and a_{21}= -6 which means a_{12}= −a_{21}. 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 + A′ is a symmetric matrix and A – A′ is a skew-symmetric matrix.

**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

Hence, 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 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.

Since for any matrix A, (kA)′ = kA′, it follows that 1 / 2 (A+A′) is a symmetric matrix and 1 / 2 (A − A′) is a skew-symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

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## Solved Examples for You

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

**Symmetric****Skew – Symmetric****Diagonal****Triangular**

**Answer :** Given A and B are Symmetric Matrices

⇒ A^{T} = A and BT = B

Now, take (ABA)^{T}

(ABA)^{T} = A^{T}B^{T}A^{T}

(ABA)^{T} = ABA

Hence, ABA is also Symmetric

**Question 2: Say true or false: If A & B are symmetric matrices of same order then AB − BA is symmetric.**

**True****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, (AB – BA)’ = – (AB – BA)

Thus, (AB – BA) is a skew-symmetric matrix

**Question 3: 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, it makes sense to discuss them being positive or negative.

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