July 3, 2013

## How to solve simultaneous linear equations using the inverse of a matrix if a determinant is zero?

I’m not even sure if this problem is technically solvable, but we’re given 4 simultaneous equations with 4 variables. We have to put these equations in matrix form (did that, used the coefficients), determine the determinant, construct the adjoint and inverse matrices, and then finally solve for the unknowns with the help of the inverse. I found the determinant to be 0 (used a program online and my graphing calc.), I found the adjoint (online program), and tried to find the inverse (calculator and program), for which there was no inverse. I know there is no inverse if the determinant is zero (the matrix is singular), but does this mean that there is no way to solve for the unknowns? I’m just not sure if there’s one little thing I’m missing (and yes I checked the initial matrix like 5 times, I even solved this way with the calculator and program using the example problem from class), or if I’m just supposed to explain to the teacher that half of this homework problem can’t technically be solved (he really doesn’t give us homework problems like this) and explain why. Here’s the equations also:

X1 + (2)X2 + X3 + X4 = 2
(2)X1 – X2 + (3)X3 + X4 = 10
(3)X1 + X2 + X3 – X4 = -3
X1 + (2)X2 – X3 – X4 = 5
So my initital matrix should be
1-2-1-1
2-(-1)-3-1
3-1-1-(-1)
1-2-(-1)-(-1)

Since your determinant is exactly 0 there is no inverse matrix so you sure aren’t going to solve the system by using it. You will either have a vector space of solutions (i/e. Infinitely many) or no solutions at all. Suggest you perform row operations on this augmented matrix trying to produce a row with the first 4 entries = 0. If the last entry is not 0, you will have no solutions.

## How do I get the inverse of the folloing demand functions?

For a duopoly i am given these two demand functions, how do i get the inverse?

x1 = 1 – p1 + ?p2

and

x2 = 1 + ?p1 – p2

The inverse demand function is basically a function which maps out quantity based on price.
So the demand function is where the quantity demanded is a function of the price. Q = f(P). So the inverse demand function is just P = f(Q)
So all you have to do is solve that equation for P. I cant really tell without more details about the modely you are using (bertrand, cournout etc..) as to what the actuall definate answer will be as they have different outputs and prices…
But they should look something like
(X1 + P1 -1 ) / alpha = P2
(X2 + P2 -1) / alpha = P1

But you should check the model as you may have to make x1 = x2 etc…
-Ben

## How to find the equation of the exponential curve that passes through the points given?

(HINT: Use the point-ratio formula y=y1*b^x-x1

a) (5,100) and (7,157)

100 * a^(x-5) will give (5, 100)

Use the second point to solve for a

157 = 100 * a^(7-5)

157/100 = a^2

Then one equation is

y = 100 * (sqrt[157]/10)^(x-5)

## How Much Would it Cost To Ship My Painting?

I have a paining on a canvas with the dimensions of 18″x24″x1.5″ and it weighs about 3 pounds. About how much would it cost for standard shipping of this painting to anywhere else in the US? Where can I go for a cheap and reliable rate? And what options do I have for shipping? ((I would appreciate any other tips as well.))

You can ship something that small pretty cheaply, like \$10. Wrap it in bubble wrap, then put two pieces of plywood/particle board/masonite on either side to prevent denting, and wrap the whole thing with tape a few times. Box it up by cutting cardboard down to fit and taping the whole thing up.

I’ve sent a lot of work this way via UPS, FedEx, and the regular old post office and never had any big problems (once, a frame cracked, but no damage was done to the painting). Absolutely insure it for more than enough to keep you covered in the worst possible situation. Also, make sure you have information about the address you’re sending it to. Is there a doorman? Will the package be left out if you don’t pay for delivery confirmation? Is it in a shady neighborhood? If no one’s home, will it be left out in the rain? Most offices, galleries, and museums are well equipped to deal with sensitive deliveries, but when sending something to a residence, it’s always good to use a little extra care and get as much information from the recipient as possible.

Rose

http://effartblog.blogspot.com

http://www.rosebriccetti.com