Fundamentals of Engineering Analysis
EGR 1302 - Introduction to Matrices
Slide 1 © 2005 Baylor University

y
Linear Systems y y=mx+b x y – mx = b rearranged to be ax + by = d
OR: ax +by+cz = d z a
1 x
1
+ b
1 x
2
+ c
1 x
3
+ d
1 x
4
= e
1 a
2 x
1
+ b
2 x
2
+ c
2 x
3
+ d
2 x
4
= e
2 a
3 x
1
+ b
3 x
2
+ c
3 x
3
+ d
3 x
4
= e
3 a
4 x
1
+ b
4 x
2
+ c
4 x
3
+ d
4 x
4
= e
4
Slide 2 © 2005 Baylor University x

How Do We Manage Large Amounts of Data?
Slide 3 © 2005 Baylor University
Matrix Algebra
We arrange data in a:
Matrix = Table = Array
The key is learning the
Definitions
Symbology
Notation

Basics of Matrix Notation
Denoted by Capital Letters A, B, C …
A =
1 2 3 4 5 6
7 8 9 5 4 3
9 8 7 6 5 4
3 5 7 2 4 6
A Matrix is referred to by
Row first, then column.
Row Column m = # rows n = # columns
This matrix A is a 4x6
A is an “m by n” or “m x n” matrix
4x6 is the “Order” of the matrix
Slide 4 © 2005 Baylor University

A =
1 2 3 4 5 5
3 4 1 7 8 3
5 6 0 2 1 7
9 8 7 6 5 4
0 3 5 7 4 2
Elements of a Matrix
Each element is denoted by lower case a ij i row, j column a
11 = 1
B =
1 2 3 4 5 6
7 8 9 5 4 3
9 8 7 6 5 4
3 5 7 2 4 6 b
34 = 6
Slide 5 © 2005 Baylor University

Order of Matrices
A= [3]
1x1 a scalar B= [1 2 3 4] a row matrix C=
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 a
21
= 2 a
12
= 1
Slide 7 © 2005 Baylor University

Slide 8 © 2005 Baylor University
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 a ij
= b ij
A = a b c d
B =
2 x
4 z
When A = B a = 2 b = x c = 4 d = z

Basic Rules of Matrices (cont.)
2. Multiply a matrix by a constant
Given k*A, where k = 2, and A =
-3 2
1 4
2A =
-6 4
2 8
Factoring: if C =
5 10
15 20
Also C = 5 *
1 2
3 4
Is not “C”!
Slide 9 © 2005 Baylor University

Slide 10 © 2005 Baylor University
Basic Rules of Matrices (cont.)
3. The Null Matrix
- All elements are Zero
A =
0 0
0 0
A is a Null Matrix

Basic Rules of Matrices (cont.)
4. Adding and Subtracting Matrices
- Must be of the same Order
A + B = C, only if
A is a “m x n” and B is a “m x n” then C is a “m x n”
A =
2 -1
3 6 a ij
+ b ij
= c ij
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

Slide 12 © 2005 Baylor University
Basic Rules of Matrices (cont.)
5. Associative Law
(A + B) + C = A + (B + C) k*(A + B) = k*A + k*B
Now for a review of this lesson -

Slide 13 © 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 ij
A = a
11 a
21 a
12 a
22

Slide 14 © 2005 Baylor University
Questions?