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 = (a_{ij})_{mxn}
A matrix A = (a_{ij})_{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 a_{ij} = 0 " i.j It is denoted by 0. 
A diagonal matrix  if m = n and a_{ij} =0 for i ¹ j 
A scalar matrix  if m = n and a_{ij} = 0 for i ¹ j = k for i ¹ j i.e. a_{11} = a_{22} ….. = a_{nn} = k (cons.) 
Identity or unit matrix  if m = n and a_{ij} = 0 for i ¹ j = 1 for i=j 
Upper Triangular matrix  if m = n and a_{ij} =0 i > j 
Lower Triangular matrix  if m=n and a_{ij} = 0 for i < j 
Symmetric matrix  if m = n and a_{ij} = a_{ij} for all I, j or A^{T} = A 
Skew symmetric matrix  if m = n and a_{ij} = a_{ij} " i,j or A^{T} = 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 A^{T}
trace I_{n} = n when I_{n} 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 AB) 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(a_{ij})_{mxn} = (Ka)_{mxn} where K is constant.
Properties:
(i) K(A+B) = KA + KB
(ii) (K_{1} k_{2})A = K_{1}(K_{2}A)=K_{2}(K_{1}A)
(iii) (K_{1} + K_{2})A = K_{1}A + K_{2}A
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. A^{m+1} = A^{m} A = AA^{m}
8 Transpose of a matrix:
A’ or A^{T} is obtained by interchanging rows into columns or columns into rows
Properties:
(i) (A^{T})^{T} = A
(ii) (A ± B)^{T} = A^{T} ± B^{T}
(iii) (AB)^{T} = B^{T}A^{T}
(iv) (KA)^{T} = KA^{T}
(v) I^{T} = I
9 Some special cases of square matrices: A square matrix is called
(i) Orthogonal matrix : if AA^{T} = I_{n} = A^{T}A
(ii) Idempotent matrix: if A^{2} = A
(iii) Involuntary matrix : if A^{2} = I or A^{1} = A
(iv) Nilpotent matrix : if $ p Î N such that A^{p} = 0
(v) Hermit an matrix : if A^{q} = A i.e. a_{ij} = `a_{ji}
(vi) Skew Hermit an matrix: if A =  A^{q}
Determinant:
1 minor & cofactor: If A = (a_{ij})_{3 x 3} then minor of a_{11} is
cofactor of an element a_{ij} is denoted by C_{ij} or F_{ij} and equal to (1)^{i+j} M_{ij}
or C_{ij} = M_{ij}, if i = j = M_{ij}, if i ¹ j
Note :
A = a_{11} F_{11} + a_{12} F_{12} + a_{13} F_{13} and a_{11} F_{21} + a_{12} F_{22} + a_{13} F_{23} = 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) A^{T} = 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=K^{n} 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. I_{n} = 1, where I_{n} is a unit matrix of order n.
(f) A zero matrix is zero i.e. 0_{n} = 0 where 0_{n} 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 (C_{ij})^{T} , where C_{ij} is cofactor of a _{ij}
Properties:
(i) A(adj A) = (adjA) A = AI_{n}
(ii) adj A = A^{n1}
(iii) ( adj AB) = (adjB)(adjA)
(iv) (adj A^{T}) = (adjA)^{T}
(v) adj(adjA) = A^{n2}
(vi) (adj KA) = K^{n1}(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^{1}A = I_{n} = AA^{1}
(iv) (A^{T})^{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,
X_{i} = (det A_{i} / det A) where A_{i} is the matrix obtained form A by replacing i^{th} column with B.
Inconsistent (with no solution) if A = 0 and at least one of the det (A_{i}) is non zero Consistent (with infinite many solution) , if A = 0 and all det (A_{i}) 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 x_{i} = 0 is called trivial solution.
(ii) Consistent (with infinitely many solution), if A = 0
(a) A = A_{i} = 0 (for determinant method)
(b) A = 0 (adj A) B = 0 (for matrix method)
Note:
A homogeneous system of equation is never inconsistent.

