Cooperation and technology transfer in pavement E ngineering
F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Inverse of Matrix. Gauss-Jordan...
-
Upload
thomasina-davidson -
Category
Documents
-
view
224 -
download
0
Transcript of F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Inverse of Matrix. Gauss-Jordan...
FUNDAMENTALS OF ENGINEERING ANALYSIS
Eng. Hassan S. Migdadi
Inverse of Matrix. Gauss-Jordan EliminationPart 1
A review of the Identity
For real numbers, what is the additive identity? Zero…. Why? Because for any real number b, 0 + b = b What is the multiplicative identity? 1 … Why? Because for any real number b, 1 * b = b
Identity Matrices
The identity matrix is a square matrix (same # of rows and columns) that, when multiplied by another matrix, equals that same matrix
If A is any n x n matrix and I is the n x n Identity matrix, then A * I = A and I*A = A
Examples
The 2 x 2 Identity matrix is:
The 3 x 3 Identity matrix is:
1 0
0 1
1 0 0
0 1 0
0 0 1
•Notice any pattern?
•Most of the elements are 0, except those in the diagonal from upper left to lower right, in which every element is 1!
Inverse review
Recall that we defined the inverse of a real number b to be a real number a such that a and b combined to form the identity
For example, 3 and -3 are additive inverses since 3 + -3 = 0, the additive identity
Also, -2 and – ½ are multiplicative inverses since (-2) *(- ½ ) = 1, the multiplicative identity
Matrix Inverses Two n x n matrices are inverses of each other if their product
is the identity Not all matrices have inverses (more on this later) Often we symbolize the inverse of a matrix by writing it with
an exponent of (-1) For example, the inverse of matrix A is A-1
A * A-1 = I, the identity matrix.. Also A-1 *A = I To determine if 2 matrices are inverses, multiply them and see
if the result is the Identity matrix!
Determine whether X and Y are inverses.
Check to see if X • Y = I.
Write an equation.
Matrixmultiplication
Now find Y • X.
Matrixmultiplication
Write an equation.
Answer: Since X • Y = Y • X = I, X and Y are inverses.
Determine whether P and Q are inverses.
Check to see if P • Q = I.
Write anequation.
Matrix multiplication
Answer: Since P • Q I, they are not inverses.
Determine whether each pair of matrices are inverses.
a.
b.
Answer: no
Answer: yes
Inverse of a number
When we are talking about our natural numbers, the inverse of a number is it’s reciprocal. When we multiply a number by it’s inverse we get 1.For example:
1
13 134 0.25 1
1k k
Inverse of a matrix
What do you think we would get if we multiplied a matrix by it’s inverse? Try it on your calculator.
1A A I A matrix multiplied by its inverse always gives us an identity matrix.
Inverse of a matrix
Not all matrices have an inverse.
If the determinant of a matrix is 0,
then it has no inverse and is said to be SINGULAR.All others are said to be NON-SINGULAR
Finding Inverses 2x2
1A A I 8 10
3 4A
Let A-1 =
dc
ba
Multiplying out gives..
10
01
43
108
dc
ba
10
01
4343
108108
dbca
dbca
043
1108
ca
ca
143
0108
db
db
Can you solve these to work out A-1?
45.1
521A
So AA-1 = I