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Fundamentals of Engineering Analysis
EGR 1302 - Introduction to Matrices
Slide 1
© 2005 Baylor University
Linear Systems
y
y
y=mx+b
x
x
z
a1x1 + b1x2 + c1x3 + d1x4 = e1
y – mx = b
rearranged to be
OR: ax +by+cz = d
ax + by = d
a2x1 + b2x2 + c2x3 + d2x4 = e2
a3x1 + b3x2 + c3x3 + d3x4 = e3
a4x1 + b4x2 + c4x3 + d4x4 = e4
Slide 2
© 2005 Baylor University
How Do We Manage Large Amounts of Data?
Matrix Algebra
We arrange data in a:
Matrix = Table = Array
The key is learning the
Definitions
Symbology
Notation
Slide 3
© 2005 Baylor University
Basics of Matrix Notation
Denoted by Capital Letters A, B, C …
123456
789543
987654
357246
A=
A Matrix is referred to by
Row first, then column.
Row - Column
m = # rows
n = # columns
This matrix A is a 4x6
Slide 4
© 2005 Baylor University
A is an “m by n” or “m x n”
matrix
4x6 is the “Order” of the matrix
Elements of a Matrix
Each element is denoted by lower case
A=
B=
Slide 5
123455
341783
560217
987654
035742
123456
789543
987654
357246
© 2005 Baylor University
aij i row, j column
a11 = 1
b34
=6
Order of Matrices
C=
B= [1 2 3 4] a row matrix
A= [3]1x1 a scalar
1
2
3
4
a column matrix
A row matrix is a “1 X n”
A column matrix is a “m X 1”
B= [1 2 3 4] is a “1 X 4” row matrix
Row or column matrices are also referred to a “Vectors”
A vector has magnitude and direction:
[x,y,z]
The coordinates of a vector are represented with a matrix
Slide 6
© 2005 Baylor University
The Square Matrix
All matrices are “rectangular”, but …
When “m = n”, the matrix is
“Square” or “n X n”
A=
3 1 4
2 0 5
6 4 2
“A” is a “3 X 3” square matrixx
a21 = 2
Slide 7
© 2005 Baylor University
a12 = 1
Basic Rules of Matrices
1. Equality – two matrices are equal if
- They are both the same “order”
- All respective elements are equal
In other words
aij = bij
A=
a b
c d
B=
When A = B
a=2
b=x
c=4
d=z
Slide 8
© 2005 Baylor University
2 x
4 z
Basic Rules of Matrices (cont.)
2. Multiply a matrix by a constant
k = 2, and A =
Given k*A, where
-3 2
1 4
2A = -6 4
2 8
Factoring: if C =
Also C = 5 *
Slide 9
© 2005 Baylor University
1 2
3 4
5 10
15 20
Is not “C”!
Basic Rules of Matrices (cont.)
3. The Null Matrix
- All elements are Zero
A=
0 0
0 0
A is a Null Matrix
Slide 10
© 2005 Baylor University
Basic Rules of Matrices (cont.)
4. Adding and Subtracting Matrices
- Must be of the same Order
A+B
A is a “m x n”
= C, only if
and B is a “m x n”
then C is a “m x n”
aij + bij = cij
A=
2 -1
3 6
B=
0 3
2 -1
A+B = C =
2 2
5 5
Subtraction: (A – B) is the same as A+ (-1)*B
Slide 11
© 2005 Baylor University
Basic Rules of Matrices (cont.)
5. Associative Law
(A + B) + C
k*(A + B)
= A + (B + C)
= k*A + k*B
Now for a review of this lesson -
Slide 12
© 2005 Baylor University
Review of Matrix Rules
- Table or Array
- Capital Letters – “A”
- Rectangular or Square
- Order: m x n, or m=n is square ( n x n)
- m = #rows, n = #columns – always “row-column”
A+B
Must be same Order
A = B if all respective elements are equal, and same Order
Element denoted by lower case
A=
Slide 13
© 2005 Baylor University
a11 a12
a21 a22
aij
Questions?
Slide 14
© 2005 Baylor University