Definition
A Matrix is a rectangular array of numbers denoted by:
Examples
Equality
Two m ´ n matrices A= [aij] and B=[bij] are equal if aij = bij for all i and j. 
Sum
Two m ´ n matrices A= [aij] and B=[bij] can be added to result in C= [cij], such that
cij = aij + bij
Example
Scalar Multiple
An m ´ n matrix A= [aij] may be multiplied by a real number c to form B= [caij]
Example
Product
If A= [aij] is an m ´ n matrix and B= [bij] is an n ´ p matrix, then the product of A and B, AB=C=[cij] is defined by:
cij = aik bkj = ai1 b1j + ai2 b2j + . . . + ain bnjfor k from 1 to n, i from 1 to m, and j from 1 to p
The general form of a matrix product is:
Example
Transpose
If A= [aij] is an m ´ n matrix, then the transpose of A, A’= [a’ij], is an n ´ m matrix defined by a’ij = aji
Algebraic Properties
A + B = B + A
A + (B + C) = (A + B) + C
A(BC) = (AB)C
(A + B)C = AC + BC
r(sA) = (rs)A = s(rA)
A(rB) = r(AB)
Diagonal Matrix
A diagonal matrix D = [aij] is an n ´ n matrix where
aii = any real number
aij = 0
Example
Scalar Matrix
A scalar matrix S = [aij] is an n ´ n matrix where
aii = m (the same real number)
aij = 0
Example
Identity Matrix
An identity matrix I = [aij] is an n ´ n matrix where
aii = 1
aij = 0
Example
Symmetric Matrix
A symmetric matrix A = [aij] is an n ´ n matrix where
A’ = A
Example
Matrix Singularity
An n ´ n matrix A is nonsingular if there exists an n ´ n matrix B such that,
AB = In
Otherwise, the matrix A is singular
Example, A is nonsingular,
If A and B are n ´ n matrices, then B is the inverse of A if
AB= In
B is denoted as A-1
Thus, AA-1=In
Example, to find A-1
Exercise
Consider the following matrices,
Compute
1- C + E
2- AB
3- CB - D
4- A’
5- B’A’
6- A(2B)
7- B-1
8- C-1