CBSE Class 12 Maths Notes: Matrices – Special Matrices

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 A² = A.

Involuntary Matrix:

A square matrix A is said to be involuntary if A² = 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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