# HallsofIvy

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1. ## Computing inverse of a 4*4, 5*5 matrix.

Yes, it is possible to find the inverse of a 4 by 4 or 5 by 5 matrix (or a 100000 by 100000 matrix). Surely you have learned that every "non-singular" matrix (having non-zero determinant) has an inverse. Whether it is 'hard' or not depends upon what you call 'hard'. It certainly is tedious! The simplest way to find the inverse of such a matrix, I think, is to use "row reduction" to reduce the given matrix to the identity matrix while applying the same operations to the identity matrix. The reason this works is that such row operation (add a multiple of one row to another, multiply a row by a number, swap two rows) corresponds to multiplication by an "elementary matrix"- one that is created by applying that same row operation to an identity matrix. If multiplying matrix "A" by a sequence of elementary matrices gives the identity matrix, then their product is the inverse matrix to A. And multiplying the identity matrix by all of them is there product. Here is a simple four by four example: Suppose $A= \begin{bmatrix}1 & 3 0 & 0 \\ -1 & 4 & 1 & 0 \\ 2 & 3 & 2 & 1\\ 0 & 2 & 1 & 0\end{bmatrix}$. I write it "side-by-side" the identity matrix: $\begin{bmatrix}1 & 3 0 & 0 \\ -1 & 4 & 1 & 0 \\ 2 & 3 & 2 & 1\\ 0 & 2 & 1 & 0 \end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ There is already a "1" leading the first row and a "0" leading the fourth row so the "reduce" the first column I can "add the first row to the second row" and "subtract twice the first row from the third row": $\begin{bmatrix}1 & 3 0 & 0 \\ 0 & 7 & 1 & 0 \\ 0 & -3 & 2 & 1\\ 0 & 2 & 1 & 0 \end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ -3 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$ There is a "7" in "second row second column" so I need to divide that row by 7 to get a "1" there. Once I have done that I can add 3 times the new second row to the third row and subtract 2 times the new second row from the fourth row to get "0" below the "1": $\begin{bmatrix}1 & 3 0 & 0 \\ 0 & 1 & \frac{1}{7} & 0 \\ 0 & 0 & \frac{17}{7} & 1\\ 0 & 0 & \frac{5}{7} & 0 \end{bmatrix}\begin{bmatrix}1 & 0 & 0 & 0 \\ \frac{1}{7} & \frac{1}{7} & 0 & 0 \\ -\frac{18}{7} & \frac{3}{7} & 1 & 0 \\ \frac{2}{7} & \frac{2}{7} & 0 & 1\end{bmatrix}$. As I said, it is tedious (and error prone) so I am going to stop here. You can continue with the last two rows and columns yourself if you wish to see that you actually do get the inverse matrix.
2. ## Rotation/Movement of triangle

"What I would do is first determine the translation necessary to move one vertex of the first triangle to the corresponding vertex of the second triangle. Then determine the rotation necessary to map the other two vertices to their corresponding vertices. The translation and rotation can be written as a single matrix multiplication using "quaternions".
3. ## General equation of the circle

What semi-circle? You have already been told that the equation of a circle with center at (a, b) and radius r is (x- a)^2+ (y- b)^2= r^2. The equation of the semi-circle consisting of the right half of that circle is (x- a)^2+ (y- b)^2= r^2 with the restriction that x> a. Similarly, the left half is the same but with x< a, the top is with y> b and the bottom y< b.
4. ## The Pi symbol "Π" is God's name in the bible, Pi is "3.14", and Ex "3:14" draws the Pi symbol when God says "I am that Pi" (new book)

Could I just point out that "the Hebrew for 'I am'" is NOT anything like "pi"? The whole basis for this thread is untrue.
5. ## Measuring acceleration due to gravity.

Of course, the specific value depends upon the units you are using. In the "mks", "meter-kilogram-second", system, the acceleration due to gravity, g, is, on the surface of the earth, approximately 9.8 meters per second squared. In "cgs", "centimeter, gram, second, it is 980 centimeters per second squared, and in the "English" system, it is 32.2 feet per second squared. That number is only an average- even on the surface of the earth it varies from place to place, largely changing with altitude but there are places inside the earth that are denser than others so even at the same altitude g may vary slightly. And, of course, on different planets, with different masses and radii, g is different. You may be confusing "g" with "Newton's universal gravitational constant, "G", used in his formula, $F= \frac{GMm}{r^2}$, for the gravitational force between two objects of masses M and m with distance r between their centers. If a mass m is on a planet with mass M and distance r from the center of the planet, then it feels gravitational force $F= \frac{GMm}{r^2}$ and, since "F= ma", would fall with acceleration $$g= a= \frac{GM}{r^2}$$. The mass of the earth and the average radius of the earth is such that that works out to g= 9.8 meters per second squared.
6. ## Public Perception of Science Survey

You understand, I hope, that posting this survey on this particular web-site will give a biased, pro-science, result.
7. ## Cosine sequence

One error is that taking n= k+ 1 in cos(2nx) gives cos(2(k+ 1)x)= cos((2k+ 2)x), not cos((2k+1)x) as you have.
8. ## Multivar equation help

I assume x, y, and z must be integers. First x+ z- y= 0. If x= 0 then z- y= 0 so z= y. (0, 0, 0), (0, 1, 1), (0, 2, 2), ..., (0, 15, 15) If x= 1 then z- y= -1 so z= y- 1 (1, 1, 0), (1, 2, 1), (1, 3, 2), ..., (1, 15, 14) If x= 2 then z- y= 2 so z= y- 2 (2, 2, 0), (2, 3, 1), (2, 4, 2), ...., (2, 14, 12) continue. ,
9. ## Aromatic compounds

Yep, there's the problem! You are misspelling the "common names". If you look up "benzene", not "bezene", you will find that Its "IUPAC" name is "benzene"! If you look up "toluene", not "toulene", you will find the "IUPAC" name is "methylbenzene". And, finally if you look up "naphthalene" rather than "naphthalene", you will find that the "IUPAC" name is, again, "naphthalene". (The "systematic IUPAC name" (I confess I don't know that that means) can be "Bicyclo[4.4.0]deca-1,3,5,7,9-pentaene" or "Bicyclo[4.4.0]deca-2,4,6,8,10-pentaene").
10. ## energy transference

I would think that people huddling together for warmth is an example of "transference of body energy"!
11. ## I have difficulties understanding relativity

You say that as if they don't normally go together!
12. ## Help designing a hypothesis or research question

You've written "an annotated bibliography with tons of sources" on what? How can you research enough to have "tons of sources" if you do not yet know what you are researching?
13. ## science is subfield of philosophy

How old are you?
14. ## Help with characteristic equation

The general solution to the quadratic equation, $ax^2+ bx+ c= 0$ is NOT $\frac{-b}{2a}\pm\sqrt{b^2- 4ac}$. It is $\frac{-b\pm \sqrt{b^2- 4ac}}{2a}$.
15. ## What is a vacuum?

Well, that sort of thing happens with every measurement. If you put a thermometer in a bowl of water to measure its temperature, you change the temperature. When you put an air-gauge on a tire to measure its air pressure, you let some air out so change the air pressure by measuring it.