## Special Matrices

Symmetric and Skew MatricesA square matrix A = [a

_{ij}] is said to be symmetric when a_{ij}= a_{ij}for all i and j. If a_{ij}= -a_{ij}for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.

For example:is a symmetric matrix and is a skew-symmetric matrix.

Hermitian and Skew - Hermitian MatricesA square matrix A = [a

_{ij}] is said to be Hermitian matrix if a_{ij}=_{ij}, i, j i.e. A^{θ}= -A.

For example:are Hermitian matrices.

Note:

* If A is a Hermitian matrix then a_{ii}=_{ij}is a real i. Thus every diagonal element of a Hermitian Matrix must be real.

* A Hermitian matrix over the set of real numbers is actually a real symmetric matrix.And a square matrix, A = [a

_{ij}] is said to be a skew-Hermitian if a_{ij}=

-_{ij}, i, j i.e. A^{θ}= -A.

For example:are skew-Hermitian matrices.

Note:

* If A is a skew-Hermitian matrix then a_{ii}=_{ij}=> a_{ii}+_{ii}= 0, i.e. a_{ii}must be purely imaginary or zero.

* A skew-Hermitian Matrix over the set of real numbers is actually a real skew- symmetric matrix.

Singular and Non-singular MatricesAny square matrix A is said to be non-singular if |A| ≠ 0, and a square matrix A is said to be singular if |A| = 0. Here |A| (or det(A) or simply det A) means corresponding determinants of square matrix A e.g. if

10 - 12 = -2 => A is a non-singular matrix.

Unitary MatrixA square matrix is said to be unitary if A = I. Since || = |A| and |A|=||, we have |||A| = 1.

Thus the determinant of a unitary matrix is of unit modulus. For a matrix to be unitary it must be non-singular.

Hence, A = I => A = I.

Orthogonal MatrixAny square matrix A of order n is said to be orthogonal if AA' = A'A = I

_{n}.

Idempotent MatrixA square matrix A is called idempotent provided it satisfies the relation A

^{2}= A.

For example:The matrix A = is idempotent as

A

^{2}= A.A = = A.

Involuntary MatrixA matrix such that A

^{2}= I is called involuntary matrix.

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

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

Illustration:Suppose a, b, c are real numbers such that abc = 1. If A = is such that A'A = I, then find the value of a

^{3}+ b^{3}+ c^{3}.

Solution:

Note A' = A.Thus, I = A'A = AA = A

^{2}|A

^{2}| = |A^{2}| = |I| = 1|A| = + 1. But |A| = a

^{3}+ b^{3}+ c^{3}- 3abc.Thus, a

^{3}+ b^{3}+ c^{3}- 3abc = + 1=> a

^{3}+ b^{3}+ c^{3}= 4, 2.

Illustration:If ω ≠ 1 is a cube root of unity, then show that

A =is singular matrix.

Solution:

Hence A is singular matrix.

Illustration:Show that the matrix = is nilpotent matrix of index 3.

Solution:

=> A

^{3}= 0 i.e. A^{k}= 0. Here k = 3.Hence A is nilpotent matrix of index 3.

Adjoint of a Square MatrixLet A = [a

_{ij}] be a square matrix of order n and let C_{ij}be cofactor of a_{ij}in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.

where C

_{ij}denotes the cofactor of a_{ij}in A.

For example:A = , C_{11}= s, C_{12}= -r, C_{21}= -q, C_{22}= p=> adj A = .

Theorem:Let A be a square matrix of order n. Then A(adj A) = |A| I_{n}= (adj A)A.

Proof:Let A = [a_{ij}], and let C_{ij}be cofactor of a_{ij}in A. Then(adj A)

_{ij}= C_{ij}1 < i, j , n.Now, (A(adj A))

_{ij}= ∑^{n}_{r=1}(A)_{ir}(adj A)_{rj}= ∑^{n}_{r=1}a_{ij}C_{jr}=> A (adj A) = = |A| I

_{n}.Similarly ((adj A)A)

_{ij}= ∑^{n}_{r=1}(adj A)_{ir}(A)_{rj}= ∑^{n}_{r=1}C_{ri}a_{rj}=Hence, A(adj A) = |A| I

_{n}= (adj A)A.

Note :The adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing the signs of off-diagonal (left hand side lower corner to right hand side upper corner) elements.

Inverse of a MatrixA non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I

_{n}= BA.In such a case, we say that the inverse of A is B and we write, A

^{-1}= B.The inverse of A is given by A

^{-1}= 1/|A|. adj A.

Properties of Inverse of a Matrix(i) Every invertible matrix possesses a unique inverse.

(ii) (Reversal law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)

^{-1}= B^{-1}A^{-1}.In general,if A,B,C,...are invertible matrices then (ABC....)

^{-1}=..... C^{-1}B^{-1}A^{-1}.(iii) If A is an invertible square matrix, then A

^{T}is also invertible and (A^{T})^{-1}= (A^{-1})^{T}.(iv) If A is a non-singular square matrix of order n, then |adj A| = |A|

^{n-1}.(v) If A and B are non-singular square matrices of the same order, then adj (AB) = (adj B) (adj A).

(vi) If A is an invertible square matrix, then adj(AT) = (adj A)T.

(vii) If A is a non-singular square matrix, then adj(adjA) = |A|

^{n-1}A.