**SPECIAL MATRICES**

** Symmetric and Skew Symmetric Matrices:**

A square matrix A = [a_{ij}] is said to be symmetric when a_{ij} = a_{ji} for all i and j, i.e. A = A¢. If a_{ij} = -a_{ji} for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix, i.e. A = – A’.

**Orthogonal Matrix:**

Any square matrix A of order n is said to be orthogonal if AA¢ = A¢ A = .

** Idempotent Matrix:**

A square matrix A is called idempotent provided it satisfies the relation A^{2} = A.

**Involuntary Matrix:**

A square matrix A is said to be involuntary if A^{2 }= I*.*

**Nilpotent Matrix:**

A square matrix A is called a **nilpotent matrix **if there exists a positive integer m such that

A^{m }= O, where O is a null matrix. If m is the least positive integer such that A^{m }= O, then m is called the **index **of the nilpotent matrix A.

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