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Finding the Inverse Matrix of the Fibonacci Sequence Eigenvector Matrix:

Updated on April 4, 2013


1. Paragraphs 7, 8, 9, 10 and 11 have a message from God.


2. This was explained at hub#12.2( linked), but it is much easier to do with a 2 X 2 matrix rather than a 4 X 4 matrix. Hub#12.2 also explained the row manipulations and what the symbols mean. The reason this hub does not look easier is we want to maintain the exact values of all the elements so that when we finally get to determining a formula for the nth term of the Fibonacci sequence then anyone can use it to any accuracy he/she wants. There is a whole lot of "rationalizing the denominator" in this hub, so if you need practice at that then you came to the right place.


3. We want element a1,1 of L1to be a negative 1 so when it is added to a2,1 then a2,1 becomes zero; therefore, a1,1 is multiplied by its negative reciprocal at L2. We continue with the instructions at the right side of L1 to get a value for a2,2 at L3, and the new value for a2,3 at L4, and for a2,4 at L5. We summarize the result of this work with the matrix at L6, and give new instructions on the right side of L6. MF1, MF2, MF3, and MF4 means multiplying factors 1,2,3, and 4. This helps to keep the work more organized.


4. We follow through with the instructions at the right side of L6 using the 2nd and 3rd MF( multiplying factors), and we get new values of all the elements for both rows one and two, ending at L14. The result of this work is summarized with the matrix at L15, and further instructions are given at the right side of L15. The work in this hub is much more organized than that of hub#12.2; therefore--in my opinion--it requires much less explanation. You just have to carefully read each image to make sense of it. Using the 4th multiplying factor at the right of L15 we get new values for the elements of row 1, and this ends at L19.1. This work is summarized with the matrix at L20. At L20 we have successfully moved the identity matrix to the left, but now we want to rationalize all the denominators in that matrix.


5. I could only find one hub here at Hub Pages that addresses rationalizing the denominator. That is catman3000's hub( linked). In addition to this hub, Khan Academy( linked) is always a very good site to access for learning math. The internet has copious instructions on how to rationalize the denominator. Anyway, we rationalize the denominators at L20, and have our final answer at L25. L25 is the inverse matrix of the Fibonacci sequence eigenvector matrix.


6. Just two more hubs are left. At the end of the last hub we will find the millionth and trillionth term of the Fibonacci sequence.


7. The choice before us is to believe God's Word, the Standard( paragraph 12 of DE, pt2 linked), or not believe it. People will say, "There is no way I will believe in a god who allows all the suffering that occurs in the world today." "If God is so powerful, and so omnipotent, then why does't He end the poverty, sickness, catastrophe, and agony in our society? "God has done nothing for me; I'll do nothing for Him, especially worship." Instead of us whining like a bunch of crybabies, let's see what God has to say about the subject. Before we stupidly, ignorantly and blasphemously point our whimpering finger at the Creator, let's see if we can figure out why our other three fingers are pointing right back at us. Sniffling, Whining, complaining, and chanting "Woe is me" does not solve problems. Resolving to solve problems, solves problems, but if the challenges become overwhelming then it is time to---FINALLY---say to God, OK, what is the message you want me to get?


8. I think in the case of sin, when we get the message, the mystery will be solved. God has given us the message thousands of years ago. If individuals and nations would have embraced the messages when God gave them to us, then a whole lot of agony throughout history would have been avoided. Here is the message, but we must stop and listen; after all it is hard to hear God while screaming at Him at the same time. We must stop whining long enough to get it: God said because they will forsake me, and break my covenant my anger will be kindled, I will forsake them, hide my face, and they shall be devoured, and many evils and troubles will befall them( Deuteronomy 31: 16 and 17); "And the Lord plagued the people because they made the calf,"( Exodus 32: 35).

9. In Leviticus, chapter 26, God could not make it any clearer than this:

"If you walk in my statutes, and keep my commandments, and do them (V3): then I will give you rain in due season, and the land shall yield her increase, and the trees of the field shall yield their fruit( V4). And your threshing shall reach unto the vintage, and the vintage shall reach unto the sowing time: and ye shall eat your bread to the full, and dwell in your land safely( V5 ). And I will give peace in the land, and you shall lie down, and none shall make you afraid: and I will rid evil beasts out of the land, neither shall the sword go through your land( V6)."

10. The next 7 verses continue to describe God's blessings upon the nation who obeys Him. Then we come to verse 14, which begins with a "But." We all know something is up with a "but," and boy is it ever!

"But if you will not hearken unto me, and will not do all these commandments"( V14); And if you shall despise my statutes, or if your soul abhor my judgments, so that you will not do all my commandments, but that you break my covenant( V15): I also will do this unto you; I will even appoint over you terror, consumption, and the burning ague, that shall consume the eyes, and cause sorrow of heart: and you shall sow your seed in vain, for your enemies shall eat it.( V16). And I will set my face against you, and you shall be slain before your enemies: they that hate you shall reign over you; and you shall flee when none pursueth you.( V17). And if you will not yet for all this hearken unto me, then I will punish you seven times more for you sins( V18)."


11. Throughout history to the present, the kind of poverty, sickness, catastrophe, sorrow, and horror that is described in this chapter( 26) of Leviticus has occurred throughout the world. After God's warnings at verses 14, 15 and 18, He warns them again in verses 21, 23 and 27, and it just keeps getting worse: heaven as iron, earth as brass; strength in vain; land barren; trees with no fruit; destruction of their children, cattle, and themselves. It even got to the point they would eat their children( V29), and because they did precisely what God warned them not to do, this prophesy came true at 2 Kings 6: 28 and 29; Jeremiah 19: 9; Ezekiel 5: 10---they ate their children. Babies scream at the top of their lungs to get what they want, and they have no intention to meet any conditions to get it. That's fine; they are babies, but it is pathetic when adults do the same thing. When atheists and apostate christians make their rants against God for all the "evil" throughout the world, for which we are directly and solely responsible, then I'm thinking we have some more aspirants for diapers. Obtuse crybabies with as much sense as a newborn baby, but having none of the innocence.


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    • Caleb DRC profile image

      Caleb DRC 4 years ago


      I agree completely. Thank you for dropping by.

      Your hubs are very beneficial and we need to get them ALL on your profile. I'll tell you how here, but to be on the safe side I'll do the same at your site.

      1. Go to your profile, and click "edit profile." 2. Scroll all the way down to "Show only featured hubs on my profile," and click "NO". 3. Go down to "Save changes", and click that. After you do that all your hubs will be on your profile.

    • FSlovenec profile image

      Frank Slovenec 4 years ago from San Francisco, CA

      God has given us everything and still people will dwell on the negative and more important not take responsibility for their actions (sins) God is God and I am not...Thank you

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Hello Newenglandsun,

      Yes I saw that and I appreciate your reply. Tomorrow is Sunday, well, let's see, it looks like we are about 47 minutes into Sunday, which is a good day for me to catch up on replies. So I'll be over there to reply to your reply.

      This is contingent on my friend not being in the dog-house, and needing to stay with me. You see his wife said, "What I want for our 30th wedding anniversary is something brand new, polished and can go from zero to 180 in less than 6 seconds." He bought her a bathroom scale, and the fight was on. What can I say? I just had to fit that joke in somewhere.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Hi April,

      Thank you so much for your complements. It is my hope that the information at the end of all these math hubs, and of course the other hubs, will give Christians, as yourself, additional argumentum( Logic: "An argument, demonstration, or appeal to reason") that may convince others of God's existence. April, I believe in heaven, but more importantly I believe in hell. One of the most ferocious animals God created is the wolverine. Occasionally, the circumstance may require that we have the ferocity of a wolverine when it comes to convincing others of God's existence. I do not believe obedient Christians do this primarily for God; He knows He exists. Nor do they do it for themselves; they know the Creator exists. If the effort appears productive I believe they primarily spend the time and energy to do this in the remote hope that the unbeliever chooses life.

      Your hubs are very beneficial. Keep up the good work and continue to stand stalwart in the faith.

    • profile image

      newenglandsun 4 years ago

      I responded to your question on one of April's hubs.

    • April Reynolds profile image

      April Reynolds 4 years ago from Arizona

      Good Morning Caleb! I have spent some time on several of your hubs this morning. I must humbly admit that a lot of the math is beyond me, but I greatly appreciate your ability to use your language to demonstrate how great and awesome and powerful God is. I also appreciate the thought and time you obviously put in to each of your comments, sharing what you know to be true with a touch of humor and lack of hostility to your opposers. Thank you.

    • North Wind profile image

      North Wind 4 years ago from The World (for now)

      Caleb, I figured as much when I went back to give it another read and it had disappeared entirely!

      It does seem like an elementary thing for man to do - simply listen to the words of God and believe however man has always had the problem of wanting to do things our way. Man yearns for some sort of power and will go down in flames (literally) trying to get it.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Thank you Jackdowney.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Thank you Ubaidh86.

    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Hi North Wind,

      I'm sorry for not responding to your last comment on the "Graphing a Rational Function" hub but Hub Pages unpublished it before I could reply. Perhaps I went over the top with the diaper comments. You know . . . diapers, rational functions, asymptotes---what's up with that? Maybe they thought I was implying something between the asymptote and diapers? I dun no. One thing I do know for sure---you wish you were on an asymptote when changing a diaper!