Lesson 4.7

Identity Matrix:  If you multiply a matrix by an identity matrix (I) the result is the same as the original matrix.  If Matrix A is a square matrix then it is also commutative.

Ones on the principal diagonal and all the rest are zeros.

Inverse Matrix  If two inverse matrices are multiplied the resulting product is the Identity Matrix.

For a 2x2 inverse matrix For example:

What about inverse matrices of other sizes?  Use calculator for now.  If you go on in math, you will learn that in Linear Algebra.

Using Inverse Matrices for Encryption and Decryption  Basic phrase: Bulldogs  Translate to a numerical string where A=1, B=2, etc.   Put this string into a matrix, maybe a 4x2 [ 0N14 A1O15 B2P16 C3Q17 D4R18 E5S19 F6T20 G7U21 H8V22 I9W23 J10X24 K11Y25 L12Z26 M13

 Multiply this matrix by another random matrix, in this case, if we do right hand multiplication, a simple 2x2 would work.  For example.

This matrix is converted back to a string of numbers To get back to the original message, the receiver would then use the inverse of our encryption matrix to get it back.

[ 0N14 A1O15 B2P16 C3Q17 D4R18 E5S19 F6T20 G7U21 H8V22 I9W23 J10X24 K11Y25 L12Z26 M13 Which translates back to: Bulldogs

Work Time 10 minutes.

Using Matrix Equations to solve systems of equations Can be converted to a matrix equation, where the variables are in their own matrix, like this: Coefficient matrix ∙ variable matrix = constant matrix

Use inverse matrices to solve: Multiply both sides by A -1 because A -1  A gives the identity.

Try this one: First, turn it into a matrix equation Then label A and B Then multiply both sides by the inverse of A. (since it is a 3x3, use calculator to do calculation.) Make sure you check your solution.