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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 3×2 and its transpose A’ is a matrix of order 2×3.
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:
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It should be a square matrix.
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After transposing the matrix, it remains the same as that of the original matrix.
Symmetric Matrix Properties
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The addition or subtraction of any two symmetric matrices will also be symmetric in nature.
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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).
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The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.
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The inverse of a symmetric matrix is also asymmetric.
Difference Between Symmetric and Skew-Symmetric Matrix
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Symmetric Matrix |
Skew-symmetric Matrix |
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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 |
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The main diagonal elements of a skew-symmetric matrix are not zero. |
The main diagonal elements of a skew-symmetric matrix are zero. |
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Symmetric Matrix Example: uploaded soon)
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Skew symmetric Matrix Example: 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