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Home » Math Homework Help » Algebra Homework Help » Symmetric/Skew Symm. Matrices
Symmetric/Skew Symm. Matrices
Symmetric Matrix: A square matrix A = [aij] is said to be symmetric if A’ = A, i.e. the transpose of a matrix is equal to the matrix itself.

In other words, a matrix A = [aij] is said to be symmetric if aij = aji, for all values of i and j.

For example, the matrices



Note: The reader should keep in mind that a rectangular matrix can never be a symmetric matrix.

Skew-Symmetric Matrix:
A square matrix A = [aij] is said to be symmetric or alternate matrix if A’ = - A.

i.e. aji = - aij for all values of i and j                       (1)

Putting j = i in (1), we get

aii = aii

i.e. 2aii = 0

or aii = 0 for all i.

This means that all the diagonal elements of a skew-symmetric matrix are zero.

The matrices



are skew-symmetric matrices.

Remark: A rectangular matrix can never be a skew-symmetric matrix.

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