Symmetric Matrix and Skew Symmetric Matrix
Symmetric Matrix
As we know that similar matrices have similar dimensions, therefore only the square matrices can either be symmetric or skewsymmetric. In other words, we can say that both a symmetric matrix and a skewsymmetric matrix are square matrices. The difference between both symmetric matrix and a skewsymmetric matrix is that symmetric matrix is always equivalent to its transpose whereas skewsymmetric matrix is a matrix whose transpose is always equivalent to its negative. For example, If M is a symmetric matrix then M = MT and if M is a skewsymmetric matrix then M = – MT
The sum of symmetric matrix and skewsymmetric matrix is always a square matrix. Below mentioned formula will be used to find the sum of the symmetric matrix and skewsymmetric matrix. For example,
Let M be the square matrix then,

M = (½) × ( M + M’) + (½) ×( M – M’)

M’ is the transpose of a matrix.

1/2( M + M’) is a symmetric matrix

1/2( M – M’) is a skewsymmetric matrix
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Symmetric Matrix Meaning
A matrix is onlys stated as a symmetric matrix if its transpose is equivalent to the matrix itself. Only a square matrix is a symmetric matrix because in linear algebra similar matrices have similar dimensions. How will you find whether the matrix given is a symmetric matrix or not ?
Generally, symmetric matrix is expressed as
M= MT
Where M is any matrix and MT is the transpose of matrix.
If aij represents any elements in an ith column and jth rows, then symmetric matrix is expressed as
aij = aji
Where each element of a symmetric matrix is symmetric in terms to the main diagonal. Let us understand the concept of symmetric matrix through the symmetric matrix example given below.
Symmetric Matrix Example
The below symmetric example helps you to clearly understand the concept of skew matrix
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In the above symmetric matrix example given below, we can see aij =aji for all the values of i and j. Here a12 = a21 = 3, = a13 = a31 = 8 a23 = a32 = 4.In other words, it is stated that the transpose of matrix M is equivalent to the matrix itself (M=MT )which implies that matrix M is symmetric.
What Is a SkewSymmetric Matrix With an Example?
A square Matrix A is defined as skewsymmetric if aij = aji for all the values of i and j. In other words, we can say that matrix P is said to be skewsymmetric if the transpose of matrix A is equal to the negative of Matrix A i.e (AT = −A). Also, it is important to note that all the elements present in the main diagonal of the skewsymmetric matrix are always zero. Let us understand this through a skewsymmetric matrix example.
SkewSymmetric Matrix Example
The below skew symmetric example helps you to clearly understand the concept of skew matrix.
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In the above skew matrix symmetric example, we can see all the elements present in the main diagonal of matrices A are zero and also a12 = 2 and a21 = 2 which implies that a12 = a21 .This conditions is valid for each values of i and j.
Properties of SkewSymmetric Matrix
Some of the properties of skewsymmetric matrix examples are given below:

When two skewmatrices are added, then the resultant matrix will always be a skewmatrix.

The result of the scalar product of skewsymmetric matrices is always a skewsymmetric matrix.

All the elements included in the main diagonal of the skew matrix are always equal to zero. Hence, the total of all the elements of the skew matrix in the main diagonal is zero.

When both identity matrix and skewsymmetric matrix are added, the matrix obtained is invertible.

The determinants of skewsymmetric matrices are always nonnegative.
Solved Example

For the Given Below Matrix M, Verify That (M + M’) Is a Symmetric Matrix.
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Solution:
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As, (M + M’) = M + M’
Hence, (M + M’) is a symmetric matrix.

Show That Matrix M Given Below is a Skew Symmetric Matrix.
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Solution:
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∴, M = M’
Hence, M is a skewsymmetric matrix.
Quiz Time

If Matrix M Is Both a Symmetric Matrix and SkewSymmetric Matrix Then Matrix M is


A scalar matrix

A diagonal matrix

A zero matrix of order m * m

A rectangular matrix

2. The Diagonal Entities of a SkewSymmetric Matrix Are


All zeroes

Can be any number

Are all equals to similar scalar k (0

None of these

1. What Are the Properties of a Symmetric Matrix?
There are multiple applications of symmetric matrices due to its properties. Some of the symmetric matrix properties are mentioned below:

It is necessary for a symmetric matrix to be a square matrix.

The eigenvalue of the symmetric matrix should always be given in a real number.

If the matrix given is invertible, then the inverse matrix will be considered a symmetric matrix

The inverse matrix will always be equivalent to the inverse of a transpose matrix.

If P and Q are symmetric matrices of equal size, then the total of (P + Q) and subtraction of (P Q) of the symmetric matrix will also be the symmetric matrix.

A scalar multiple of a symmetric matrix will also be considered as a symmetric matrix.

If the symmetric matrix has different eigenvalues, then the matrix can be changed into a diagonal matrix. In other words, a symmetric matrix is always diagonalizable.

Eigenvectors are orthogonal for every different eigenvalue,
2. What Are the Determinants of a SkewSymmetric Matrix?
If M is a skewsymmetric matrix, which is also considered as a square matrix, then the determinant of M should satisfy the belowgiven situation:
Det (MT) = det (M) = (1)n det(M)
The inverse of skewsymmetric matrix is not possible as the determinant of it having odd order is zero and therefore it is singular.
The determinants of a skewsymmetric matrix is also one of the properties of skewsymmetric matrices. If we have any skewsymmetric matrix with odd order then we can straightly write its determinants equals to zero. The property of the determinants of a skewsymmetric matrix can be verified using an example of 3 by 3 matrix. The determinants of a skew matrix can be found out using cofactors and can state that its determinant is equivalent to zero.