Matrices:
1 Matrix- A system or set of elements arranged in a rectangular form of array is called a matrix.
2 Order of matrix: If a matrix A has m rows & n columns then A is of order m x n .
The number of rows is written first and then number of columns . Horizontal line is row & vertical line is columns
3 Types of matrices : A matrix A = (aij)mxn
A matrix A = (aij)mxn over the field of cimplex number is said to be
| Name | Properties |
| A row matrix | If m =1 |
| A column matrix | if n =1 |
| rectangular matrix | if m ¹ n |
| A square matrix | if m = n |
| A null or zero matrix | if aij = 0 " i.j It is denoted by 0. |
| A diagonal matrix | if m = n and aij =0 for i ¹ j |
| A scalar matrix | if m = n and aij = 0 for i ¹ j
= k for i ¹ j
i.e. a11 = a22 ….. = ann = k (cons.) |
| Identity or unit matrix | if m = n and aij = 0 for i ¹ j
= 1 for i=j |
| Upper Triangular matrix | if m = n and aij =0 i > j |
| Lower Triangular matrix | if m=n and aij = 0 for i < j |
| Symmetric matrix | if m = n and aij = aij for all I, j
or AT = A |
| Skew symmetric matrix | if m = n and aij =- aij " i,j
or AT = -A |
4 Trace of a matrix: Sum of the elements in the principal diagonal is called the trace of a matrix.
trace (A ± B) = trace A ± trace B
trace kA = k trace A
trace A = trace AT
trace In = n when In is identity matrix.
trace AB ¹ trace A trace B.
5 Addition & subtraction of matrices : If a and B are two matrices each of order same, then A+B (or A-B) is defined and is obtained by adding (or subtracting) each element of B form corresponding element of A
6 Multiplication of a matrix by a scalar:
KA = K(aij)mxn = (Ka)mxn where K is constant.
Properties:
(i) K(A+B) = KA + KB
(ii) (K1 k2)A = K1(K2A)=K2(K1A)
(iii) (K1 + K2)A = K1A + K2A
7 Multiplication of matrices: Two matrices A & B can be multiplied only if the number of columns A is same as the number of rows in B.
Properties:
(i) In general matrix multiplication is not commutative i.e. AB ¹ BA .
(ii) A(BC) = (AB)C [ Associative law]
(iii) A.(B+C) = AB + AC [ Distributive law]
(iv) If AB = AC Þ B = C
(v) If AB = 0, then it is not necessary A = 0 or B =0
(vi) AI = A = IA
(vii) ,atrix multiplication is commutative for + ve integral i.e. Am+1 = Am A = AAm
8 Transpose of a matrix:
A’ or AT is obtained by interchanging rows into columns or columns into rows
Properties:
(i) (AT)T = A
(ii) (A ± B)T = AT ± BT
(iii) (AB)T = BTAT
(iv) (KA)T = KAT
(v) IT = I
9 Some special cases of square matrices: A square matrix is called
(i) Orthogonal matrix : if AAT = In = ATA
(ii) Idempotent matrix: if A2 = A
(iii) Involuntary matrix : if A2 = I or A-1 = A
(iv) Nilpotent matrix : if $ p Î N such that Ap = 0
(v) Hermit an matrix : if Aq = A i.e. aij = `aji
(vi) Skew- Hermit an matrix: if A = - Aq
Determinant:
1 minor & cofactor: If A = (aij)3 x 3 then minor of a11 is
cofactor of an element aij is denoted by Cij or Fij and equal to (-1)i+j Mij
or Cij = Mij, if i = j =- Mij, if i ¹ j
Note :
|A| = a11 F11 + a12 F12 + a13 F13 and a11 F21 + a12 F22 + a13 F23 = 0
2 Determinant: if A is a square matrix then determinant of matrix is denoted by det A or |A|.
Expansion of determinant of order 3 x 3
Propertes:
(i) |AT| = |A|
(ii) By interchanging two rows (or columns), value of determinant differ by –ve sign.
(iii) If two rows (or columns) are identical then |A| = 0
(iv) |KA|=Kn det A, A is matrix of order n x n
(v) If same multiple of elements of any row (or column) of a determinant are added to the corrsponding elements of any other row (or column), then the value of the new determinant remain unchanged .
(vi) Determinant of :
(a) A nilpotent matrix is 0.
(b) An orthogonal matrix is 1 or -1
(c) A unitary matrix is of modulus unity.
(d) A Hermitian matrix is purely real.
(e) An identity matrix is one i.e. |In| = 1, where In is a unit matrix of order n.
(f) A zero matrix is zero i.e. |0n| = 0 where 0n is a zero matrix of order n
(g) A diagonal matrix = product of its diagonal elements.
(h) Skew symmetric matrix of odd order is zero.
3 Multiplication of two determinants:
Multiplication of two second order determinants is defined as follows.
If order is different then for their multiplication, express them firstly in the same order.
Matrices and Determinants :
1 Adjoint of a matrix:
Adj A (Cij)T , where Cij is cofactor of a ij
Properties:
(i) A(adj A) = (adjA) A = |A|In
(ii) |adj A| = |A|n-1
(iii) ( adj AB) = (adjB)(adjA)
(iv) (adj AT) = (adjA)T
(v) adj(adjA) = |A|n-2
(vi) (adj KA) = Kn-1(adj A)
2 Inverse of a matrix :
(i) A-1 exists if A is non singular i.e. |A| ¹ 0
(ii) A-1= {(adjA)/|A|}, |A| ¹ 0
(iii) A-1A = In = AA-1
(iv) (AT)-1 = (A-1)T
(v) (A-1)-1 =A
(vi) |A-1| = |A|-1 = (1/|A|)
(vii) If A & B are invertible square matrices then
(AB)-1 = B-1 A-1
3 Rank of a matrix:
A non zero matrix A is said to have rank r, if
(i) Every square sum matrix of order (r+1) or more is singular
(ii) There exists at least one square sub matrix of order r which is non singular.
4 Homogeous & non homogeneous system of linear equation :
A system of equations Ax = B is called a homogenous system if B =0 If B ¹ 0 then it is called non homogeneous system equation.
5 (a) Solution of non homogeneous system of linear equation :
(i) Cramer’s rule: Deteminant method
The non homogeneous system Ax = B, B ¹ 0 of n equation in n variable is –
Consistent (with unique solution) if |A| ¹ 0 and for each I = 1,2, …… n,
Xi = (det Ai / det A) where Ai is the matrix obtained form A by replacing ith column with B.
Inconsistent (with no solution) if |A| = 0 and at least one of the det (Ai) is non zero Consistent (with infinite many solution) , if |A| = 0 and all det (Ai) are zero.
(ii) Matrix method:
The non homogeneous system Ax = B, B ¹ 0 of n equation in n variable is –
Consistent (with unique solution) if |A| ¹ 0 i.e. if A is non singular, x = A-1 B
Inconsistent (with no solution), if |A| = 0 and (adj A) B is a non null matrix.
Consistent (with infinitely ,many solution), if |A| = 0 and )adj A) B is null matrix.
(b) Solution of homogeneous system of linear equations:
The homogeneous system Ax = B, B = 0 of n equations in n variables is
(i) Consistent (with unique solution) if |A| ¹ 0 and for each I = 1,2, …….. n xi = 0 is called trivial solution.
(ii) Consistent (with infinitely many solution), if |A| = 0
(a) |A| = |Ai| = 0 (for determinant method)
(b) |A| = 0 (adj A) B = 0 (for matrix method)
Note:
A homogeneous system of equation is never inconsistent.
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