Symmetric and Skew Matrices
A square matrix A = [aij] is said to be symmetric when aij = aij for all i and j. If aij = -aij for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix.
is a symmetric matrix and is a skew-symmetric matrix.
Hermitian and Skew - Hermitian Matrices
A square matrix A = [aij] is said to be Hermitian matrix if aij = ij, i, j i.e. Aθ = -A.
are Hermitian matrices.
* If A is a Hermitian matrix then aii =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 = [aij] is said to be a skew-Hermitian if aij =
-ij, i, j i.e. Aθ = -A.
are skew-Hermitian matrices.
* If A is a skew-Hermitian matrix then aii =ij => aii + ii = 0, i.e. aii 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 Matrices
Any 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.
A 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.
Any square matrix A of order n is said to be orthogonal if AA' = A'A = In.
A square matrix A is called idempotent provided it satisfies the relation A2 = A.
The matrix A = is idempotent as
A2 = A.A = = A.
A matrix such that A2 = I is called involuntary matrix.
A square matrix A is called a nilpotent matrix if there exists a positive integer m such that Am = O. If m is the least positive integer such that Am = O, then m is called the index of the nilpotent matrix A
Suppose a, b, c are real numbers such that abc = 1. If A = is such that A'A = I, then find the value of a3 + b3 + c3.
Note A' = A.
Thus, I = A'A = AA = A2
|A2| = |A2| = |I| = 1
|A| = + 1. But |A| = a3 + b3 + c3 - 3abc.
Thus, a3 + b3 + c3 - 3abc = + 1
=> a3 + b3 + c3 = 4, 2.
If ω ≠ 1 is a cube root of unity, then show that
A =is singular matrix.
Hence A is singular matrix.
Show that the matrix = is nilpotent matrix of index 3.
=> A3 = 0 i.e. Ak = 0. Here k = 3.
Hence A is nilpotent matrix of index 3.
Adjoint of a Square Matrix
Let A = [aij] be a square matrix of order n and let Cij be cofactor of aij 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 Cij denotes the cofactor of aij in A.
For example: A = , C11 = s, C12 = -r, C21 = -q, C22 = p
=> adj A = .
Theorem: Let A be a square matrix of order n. Then A(adj A) = |A| In = (adj A)A.
Proof: Let A = [aij], and let Cij be cofactor of aij in A. Then
(adj A)ij = Cij 1 < i, j , n.
Now, (A(adj A))ij = ∑nr=1(A)ir (adj A)rj = ∑nr=1 aij Cjr
=> A (adj A) = = |A| In.
Similarly ((adj A)A)ij= ∑nr=1(adj A)ir (A)rj= ∑nr=1Cri arj=
Hence, A(adj A) = |A| In = (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 Matrix
A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = In = 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 AT is also invertible and (AT)-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-1A.