1
Matrices
Properties, Operations, Algebra
Matrices and their properties
•
A matrix is a rectangular array of numbers; an
m x n
matrix
has
m
rows and
n
columns
m x n
has
rows and
n
columns
•
If
m = n
we call it a
square matrix
•
A
zero
matrix is one which has zero entries
everywhere
•
Two matrices are
equal
if they have the same
i
d
ll th i
t i
l
size and all their entries are equal
Matrices and their properties
•
Column matrix
: is
m x 1
•
Row matrix
: is
1 x n
•
scalar multiple of a matrix
:
kA
, where each entry is
multiplied by
k
Operations on matrices: Addition/Subtraction
•
Two matrices can be
added
or
subtracted
if they have the
same size;
A + B , A - B
•
The result is obtained just by adding or subtracting
individual entries

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Operations on matrices: Multiplication
•
The product
AB
of two matrices is defined
only
if the left
matrix,
A
, has the same number of columns as the right
matrix,
B
, has rows
•
m x r
can multiply
r x n
to give
m x n
•
Note the order matters!!
Operations on matrices: Multiplication
•
If
A
is an
m x r
matrix, and
B
is an
r x n
matrix, then the
product
A
B
= C
is the
m x n
matrix whose entries are
determined as follows:
•
to find the entry in
row i
,
column j
of
AB
–
single out
row i
from the matrix
A
and
column j
from
the matrix
B
–
multiply together the corresponding entries from the
row and column
–
add up the resulting products.
Example: Product of 2 matrices
•
A: 2 x 4,
B: 4 x 7,
C: 7 x 2
–
can give 9 potential products of which only 3 are
can give 9 potential products, of which only 3 are
defined
Matrix multiplication example
1
2
1
2
3
1
1
3
3
1
4
3
1
1
2