Transcript matrices1
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.