# Symmetric Matrix

## Symmetric and Skew Symmetric Matrix

In science, commerce, and even in our everyday life it is often convenient to represent a set of numbers in rows and columns, called arrays. Suppose a company has three factories X, Y and Z which produces four commodities A, B, C and D. Further assume that X produces 4, 0, 7 and 6 units of A, B, C, and D respectively; the corresponding units produced by Y and Z are 10, 3, 8, 5 and 8, 9, 2, 11 respectively. This information relating to production can be displayed by the following array:

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Clearly, each row represents the number of units of a particular commodity produced by three factories and each column represents the number of units of different commodities produced in a particular factory. With this sense in advance the above array can be written as follows:

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Such an array of numbers arranged in rows and columns is called a matrix. The above matric has 4 rows and three columns. This array is called a matrix of order 4 x 3.

Definition: A matrix is a rectangular array of numbers that are arranged in rows and columns. If m.n numbers are arranged in a rectangular array of m rows and n columns, it is called a matrix of order m by n (written as m x n).

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The numbers a11, a12, a13, etc constituting a matrix are called elements or entries of the matrix. The matrix is the element in the ith row and jth column. The capital letters are used to denote matrices,

### Transpose of a Matrix

Let A be a matrix of order m x n; then the matrix of order n x m obtained by interchanging the rows and columns of A is called Transpose of the matrix A and is denoted by A’ or AT. For example, if

A = [fig 6] then A’= [fig 7]

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Note that A is a matrix of order 3x2 and its transpose A’ is a matrix of order 2x3.

### Symmetric Matrix

A square matrix that is equal to its transpose is called a symmetric matrix. For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. Note that if A is a symmetric matrix then A’ = A where A’ is a transpose matrix of A.

Thus, A= [fig 4] is a symmetric matrix of order 3.

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Note that the transpose of A = A’ = [fig5] = A.

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### How to Know If a Matrix is Symmetric

To know if the given matrix is symmetric or not, check the following conditions:

1. It should be a square matrix.

2. After transposing the matrix, it remains the same as that of the original matrix.

### Symmetric Matrix Properties

1. The addition or subtraction of any two symmetric matrices will also be symmetric in nature.

2. The product of two symmetric matrices [A and B] doesn’t always give a symmetric matrix [AB]. The result of the product is symmetric only if two individual matrices commute (AB=BA).

3. The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.

4. The inverse of a symmetric matrix is also asymmetric.

## Difference Between Symmetric and Skew-Symmetric Matrix

 Symmetric Matrix Skew-symmetric Matrix Symmetric Matrix definition: Transpose of a matrix is always equal to the matrix itself.           A T= A Skew-symmetric Matrix definition: Transpose of a matrix is always equal to the negative of the matrix itself.        AT= -A The main diagonal elements of a skew-symmetric matrix are not zero. The main diagonal elements of a skew-symmetric matrix are zero. Symmetric Matrix Example: (image will be uploaded soon) Skew symmetric Matrix Example: (image will be uploaded soon)

### Determinant of Matrix

A fixed number that defines a square matrix is called the determinant of a matrix.

The process of finding the determinant of a symmetric matrix and the determinant of skew-symmetric is the same as that of a square matrix.

### Matrix Inverse of a Symmetric Matrix

If A and B are two square matrices of the same order such that AB = BA = I, where I is the unit matrix of the same order as A. or B, then either B is called the inverse of A or A is called the inverse of B. The inverse of matrix A is denoted by A-1.

The inverse of a square matrix A exists if |A| is not equal to 0.

If A is nonsingular then, A-1 = $\frac{adj(A)}{|A|}$

Let  A and B are two nonsingular Matrices then,

i) A-1. A = A. A-1 = I

ii) (A-1)-1=A

iii) (A-1)T = (AT)-1

iv) (AB)-1= B-1A-1

FAQ (Frequently Asked Questions)

Question 1: Is Every Symmetric Matrix Orthogonal?

Answer: No, not every symmetric matrix is an orthogonal matrix. We know that a matrix to be symmetric, its transpose must be equal to itself (A=AT) whereas, for a matrix to be orthogonal, its product with its orthogonal must be an Identity matrix (A. AT= I).

Thus, a symmetric matrix A is also orthogonal only if A2 = I

Question 2: What is an Orthogonal Matrix?

Answer: A square matrix of order n is said to be orthogonal if A. AT= AT. A= I where AT is the transpose of A and I is the unit matrix of order n.

If A is an orthogonal matrix then,

A. AT= AT. A= I                ---------------------- (1)

Again, by the properties of determinants we have,

Det.AT = det.A and det.A. AT= Det.A Det.AT= (Det.A)2

From eq (1) we can say that

det.A. AT= det I = 1 or (Det.A)2 = 1 or Det.A= ±1

If det A = 1, then A is called a proper orthogonal matrix.

If det A = -1, then A is called an improper orthogonal matrix.