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Symmetric/Skew Symm. Matrices

**Symmetric Matrix:**A square matrix

**A = [a**is said to be symmetric if

_{ij}]**A’ = A**, i.e. the transpose of a matrix is equal to the matrix itself.

In other words, a matrix

**A = [a**is said to be symmetric if

_{ij}]**a**

_{ij}= a_{ji}, 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.

**A square matrix**

Skew-Symmetric Matrix:

Skew-Symmetric Matrix:

**A = [a**is said to be symmetric or alternate matrix if

_{ij}]**A’ = - A**.

i.e.

**a**

_{ji}= - a_{ij}for all values of

**i**and

**j**

**(1)**

Putting

**j = i**in

**(1)**, we get

**a**

_{ii}= a_{ii}

i.e.

**2a**

_{ii}= 0or

**a**for

_{ii}= 0**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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