MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar MTH Module Notes: SYSTEMS OF EQUTONS & MTES. MT NVESES & POPETES OF MTES Definition: We cannot discuss the inverse of a matri unless we defined identit matrices. The multiplicative identit of a matri is a square matri of order n sie nn denoted b, with s along the principal diagonal top left to bottom right s everwhere else. elow are eamples of a few identit matrices. dentit Matri: dentit Matri: dentit Matri: Propert of dentit Matrices: Given a square matri the same order as ; f = =, then is called the multiplicative identit or just identit of. There is also an additive identit called the ZEO matri, such that += +=. Discussed below. Eample :, heck that This is alwas true for an matri. Eample : This is alwas true for an. Matrices have an DDTVE DENTTY but the are of little use. will give ou an eample. matri = additive inverse of a matri. vice-versa. Note bout eal Numbers: With real numbers, the multiplicative identit is, because a*=a *a=a. Warning: eal numbers matrices are different sets. The identit of a real number is a number. The identit of a matri is a matri see above. The multiplicative inverse of a real number a is /a, because a*/a = the identit. The same logic holds true for matrices. Two square matrices whose product is a matri identit are said to be multiplicative inverses or just inverses, since we have little use for an additive inverse of a matri. eal Numbers Matrices The two real numbers b a are inverses because. The matrices are inverses since &
MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar Definition: f is a square {nn} matri is another square {nn} matri, such that = =, then is said to be the inverse of likewise is also the inverse of. We denote inverse as -. The epression - is not the same as /. There is no division operation for matrices. To show that two matrices are inverses, ou must make sure the are both square matrices of the same sie. The two matrices below are square matrices of the same sie {}. Show that these matrices satisf the above definition thus are inverses of one another. / / Solution: / / / / Finding the inverse of a given square matri. Form the ugmented Matri. Use row operations to transform into. The matri = - matri = -. heck the result optional Eample: Find the inverse of =.. / / / / / / / / / / / / / / /. / /. heck: Show / Practice: Find the inverse of / nswer: Note: Do not use the calculator cheat buttons to find the inverse. lso, avoid the formula for finding the inverse of a {} matri, unless our instructor allows ou to use it. t onl works for {} case, so ou will be stuck if ou need the inverse of a {} matri.
MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar Eample : Use Gauss-Jordan Elimination to find the inverse of. Note: n this eample duplicate matrices to ease navigation. n the net eample, will not. ~ hecking: Multipl to see that, Eample : Use Gauss-Jordan Elimination to find the inverse of. ~ / / / / / / / / / / / / / / / hecking: Multipl to see that,
MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar rptograph: There are man tpes of crptograph techniques. n an earlier lecture we talked about using functions to encode decode messages. Here, we will use matrices to encode a message. To encode a message with a matri, ou must use a matri that has an inverse. You can use a {} matri, a {} matri, a {} so on. The sie of the encoding matri determines how ou enter the message into matri form. See below. lphabet ode: Let =, =, = so on. Let a blank space =. You could use an number other than - for a blank. lso, ou could write the message without using blanks. Message To Encode: SEET ODE or {,,,,,,,,,,} Encoding Matri: We will multipl this matri b the message. Encoding: When we write the message in matri form, it must be written as a {n} matri. Otherwise, multiplication of the matrices would be undefined. The best wa to do this is to let the first column be {,}, the second column be {,}, the third column be {,} so on. M The last entr is ero because we cannot leave empt To encode, we multipl M: M = Now the encoded message is: {,,,,,,,,,,,} Decoding: We would send this message to our partner. How would our partner decode the message? / / Find the inverse of. / / Now multipl this b the encoded message M. / M / / / M Now our partner has decoded the message can write out the secret message.. {,,,,,,,,,,} = SEET_ODE
Using a encoding matri. MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar SEET_ODE = {,,,,,,,,,,} Message Matri: M will use this matri to encode: M = Encoded Message: {,,,,-,,,-,,-,,} To decode the message use the inverse of from earlier eample. Practice Problem: The encoded message is {,,,,,,,,,,,,,,,,,} The above message was encoded with. Decode the message. nswer elow. Hint: nswer: WHO S L GUSS Note bout Encrption ver simple technique to encode a nother encoding technique is a transformation message - Shift pher. cpher. You pair each letter with another rom MTHEMTS can easil be letter {M-P, -J, T-G, etc}. This is more difficult to encoded as PDWKHPDWLFV crack but not for eperts. How did do this? Encrption has been used thouss of ears. From just translate each letter units. medieval das to present da. No encrption MNOP,D,TUVW, etc. scheme is % secure, but the scheme needs to be Just reverse this to get back the word secure enough so that the reward is not worth the MTHEMTS effort. This is eas to do, but also eas to Toda s computers use complicated algorithms crack. person could just shift each involving ver large prime numbers. am no epert, letter space, space, space, until but suppose ou had the number needed it s ou find something that makes sense. prime factors to decode a message. This is eas,. What if the number was,,,,,? How long would it take ou to factor it into the prime numbers,,,,? Now consider even larger numbers. Even with supercomputers this can be time consuming!
MTH Finite Mathematics or usiness Math Lecture Notes uthor / opright: Kevin Pinegar Properties of Matrices asic Properties of Matrices- ssuming that all operations are defined for the indicated matrices,,,, then the following properties hold true. dditive Properties o ssociative: ++=++ o ommutative: +=+ o dditive dentit: +=, += o dditive nverse: +-= Multiplication Properties o ssociative Properties: = o No commutative propert o Multiplicative dentit: =, = o Multiplicative nverse: - =, - = f is a square matri ombined Properties o Left Distributive Propert: +=+ o ight Distributive Propert: +=+ Equalit Properties o ddition: f =, the +=+ o Left Multiplication: f =, the = o ight Multiplication: f =, the = Solve the Matri Equations For the Matri Solving Sstems Of Equations using the Matri nverse Method Suppose ou wish to solve the following sstem b using the Matri nverse method. The sstem can be written as = = Where represents the coefficient matri, the solution matri the constant matri, The solution matri to = is = -. see above The solution is -,, NOTE: You have to find the inverse if its not given, but once ou have it ou can solve man of the same sstems where onl the constants change.