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