MATRICES

 The concept of matrices was developed by a Mathematician Arthur Cayley . Matrices are useful in expressing numerical information in compact form. They are effectively used in expressing different operators .

Definition :

     A rectangular arrangememnt of mn numbers in m rows and n columns, emclosed in [ ] or ( ) is called a matrix of order m by n .

 A matrix by itself does not have a value or any special meaning . Order of the matrix is denoted by m m🗙n , read as m by n . Each member of the matrix is called an element of the matrix . Matrices are generaly denoted by A, B, C, ... and their elements are denoted by a_{ij ,}b_ij  , c_{ij , ...etc.}

_{Matrix is generaly written as :}

Important Formulas for Matrices 

If A and B are square matrices of order n, and In is a corresponding unit matrix , then

(a) A(adj.A) = | A | In = (adj A) A

(b) | adj A | = | A |n-1 (Thus A (adj A) is always a scalar matrix)

(c) adj (adj.A) = | A |n-2 A

(d) adj (AB) = (adj B) (adj A)

(e) adj (Am) = (adj A)m,

(f) adj 0 = 0

(g) A is symmetric ⇒adj A is also symmetric

(h) A is diagonal ⇒adj A is also diagonal

(i) A is triangular ⇒adj A is also triangular

(j) A is singular ⇒| adj A | = 0

(k) | adj(adj.A)| = |A|^(n-1)(n-1)

(l) adj(k A) = K ^n-1 (adj A) , k ∈ R

(m) adj(I_n ) = I_n.

_{Types Of Matrix :}

_{1. Row Matrix:}

       _{A matrix having only one row is called as a row matrix . It is of order 1x n , where n   ≥1 . ex., i) [2  5] ii) [2  3  6] .}

_{2. Column Matrix :}

     A matrix having only one column is called as a column matrix . It is of order m x 1 , where m ≥1 .

3. Zero or Null Matrix :

  A matrix in which every element of matrix is zero is callled as a zero or null matrix . It is denoted by 0.

_{4.  Square Matrix :}

   _{A matrix with equal number of rows and column is called as a square matrix .}

_{5.  Diagonal Matrix :}

    _{A square matrix in which every non-diagonal elememt is zero ,is called as a diagonal matrix .}

Note : If a_{11 ,}b_22  , c₃₃ are diagonal element of a diagonal matrix A of order 3 , then we write the matrix A as A = Diag.

6.   Scalar Matrix :

     A diagonal matrix in which all the diagonal elements are same , is called as a scalar matrix .

7.   Unit or Identity Matrix :

     A scalar matrix in which all the diagonal elements are 1 (unity) , is callled as identity matrix of order n is denoted by I_n.

Note : i)  Every identity matrix is scalar matrix but every scalar matrix is need not be identity matrix .                  ii)  Every scalar matrix is diagonal matrix but every diagonal matrix need not be a scalar matrix.

8.   Uppper Trigular Matrix :

     A square matrix in which every element below the diagonal is zero , is callled as upper trigular matrix .

_{9.   Lower Trigular Matrix :}

    _{A square matrix in ewhich every element above the diagonal is zero , is called as lower triular matrix .}

_{10. Trigular Matrix :}

    _{A square matrix is called as trigular matrix if it is an uppper trigular or a lower trigular matrix.}

_{Note:  The diaggonal , unit and null matrices are also trigular matrix.}

_{11. Symmetric Matrix:}

          _{A square matrix A = [} a_ij]_nn   in which  a_ij  = a_ji for all i and j , is called a symmentric matrix .

12. Skew - Symmetric Matrix :

      A square matrix A= _[ a_ij]_nn  in which   a_ij  = -a_jifor all i and j , is called a skew - symmetric matrix. In skew -symmetric matrix each diagonal element is zero.

13.  Determinant of a Matrix :

    Determinant of a matrix is defined only for a square matrix . If A is a square matrix , then the same arrangement of as also gives us a determinant , by replacing square brackets by vertical bars. It s denoted by |A| or det(A) .  If A =_[ a_ij]_nn then is pf order n.

14.  SIngular Matrix :

     A square matrix A is said to be singular matrix if |A| = det(A) = 0 , otherwise it is said to be non-singualr matrix.

15. Transpose of a Matrix :

   The matrix obtained by interchanging rows and columns of matrix A is called as Trnaspose of matrix A . It is denoted by A' or  AT. 

eg., 

Properties of Transpose of Matrix

(i) (AT)T= A (ii) (A + B)T = AT+ BT (iii) (AB)T = BTAT (iv) (kA)T = k(A)T

(v) (A1A2A3 ……An-1An)T =

(vi) IT = I (vii) tr(A) = tr(AT)

16.  Hermitian and skew – Hermitian matrix:

   A = Aθ     (Hermitian matrix)(Aθ represents conjugate transpose)

    Aθ  = -A     (skew-Hermitian matrix).

17.  Orthogonal matrix: if AAT = In = ATA

18. Idempotent matrix: if A2 = A

19.  Involuntary matrix: if A2 = I or A-1 = A

20.  Nilpotent matrix: A square matrix A is nilpotent; if Ap = 0, p is an integer.

Trace of Matrix

The trace of a square matrix is the sum of the elements on the main diagonal.

(i) tr(λA_ = λ tr(A)

(ii) tr(A + B) = tr(A) + tr(B)

(iii) tr(AB) = tr(BA)

Properties of Matrix Multiplication

(i) AB ≠ BA

(ii) (AB)C = A(BC)

(iii) A.(B + C) = A.B + A.C

Adjoint of a Matrix:

Inverse of a Matrix :

Matrix Transpose :

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