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An Introduction to
Matrix Algebra
King Saud University
Matrix Operations
• Although we introduced matrices as a
structure for convenient “bookkeeping”
when solving systems of linear equations,
they are interesting mathematically in their
own right.
• We can define the operations of addition,
scalar multiplication, subtraction and
multiplication on them.
Matrix Terminology
• We say that two mxn matrices A and B are
equal if they have the same size and aij=bij
for 1≤ i ≤ m and 1 ≤ j ≤ n.
• A matrix with only one row is called a row
matrix or row vector.
• A matrix with only one column is called a
column matrix and column vector. (ai)
Matrix Addition
• If A and B are 2 mxn matrices then A+B is
the mxn matrix with entries
(a+b)ij= aij+bij.
Scalar Multiplication
• If A is an mxn matrix and c is a scalar then
cA is the mxn matrix with entries,
(ca)ij = caij.
• With this definition we can define A-B to be
A+(-1)B.
Matrix Multiplication
• If A is an mxp matrix and B is an pxn matrix
then AB is the mxn matrix with entries,
p
(ab )ij   aik bkj .
k 1
Why????
• Consider the system of equations,
a1,1 x  a1,2 y  a1, 3 z  b1
a2,1 x  a2,2 y  a2,3 z  b2
a3,1 x  a3,2 y  a3,3 z  b3
• If A is the coefficient matrix of this system
x is the column matrix of variables and b is
the column matrix of right hand side
numbers then the system is expressed as
Ax = b.
Partitioned Matrices
• Sometimes it is also useful to think of a
linear system in the following way:
 a11 
 a12 




a21 
a22 


x1
x2









 am 1 
 am 2 
 a1n   b1 

  
a2 n   b2 

xn


  

  
 amn   b m 
• We say here that we have partitioned A
into its column matrices a1, a2, ... , an, and
written b as a linear combination of a1, a2,
... , an.