Matrices, I can't seem to cancel out any more elements

In summary: So the general form of the equation is z= 6.In summary, the student is confused on how to solve a system of equations. His professor did an example where he has 3 equations and 5 unknowns, and then showed him how to solve the system using a linear combination.
  • #1
mr_coffee
1,629
1
Hello everyone, every row operation I make now is taking out the 0's i just made, any ideas on what I can do from here?
Here is my work:
http://img201.imageshack.us/img201/5573/lastscan2ah.jpg [Broken]

Also I was confused on how this works, my professor did an example:
3x+2y+z = 8;
x = a
y = b
z = 8 - 3a - 2b
I get this part but then he does the following:

http://img140.imageshack.us/img140/1337/eeeeeee1ut.jpg [Broken]

Where is he getting those values? The first one looks like he let a = 0 and b = 0, then ur left with just 8, but I'm not sure.
Thanks.
 
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  • #2
You have 3 equations and 5 unknowns, so you will have 2 free variables. You can't reduce it any more than that.
 
  • #3
hm...What do i do from there then? How can you solve the system?
 
  • #4
He's just writing it as a sum of vectors multiplied by some coefficient. For example,

3+x 3 1 0
x^2 = 0 *1 + 0 *x + 1 *x^2
X^2-4x+7 7 -4 1

or if you write it out as 3 separate equations, which it actually is,

3 + x = 3*1 + 1*x + 0*x^2
x^2 = 0*1 + 0*x + 1*x^2
x^2-4x+7 = 7*1 + -4*x + 1*x^2
 
  • #5
Why does it look like all he is doing is letting a and b equal 0 in the first one, which will make 0 0 8 then he's letting a = 1, and b = 0, and finding the values then, letting a = 0 and b = 1, and then finding the values?
 
  • #6
Because that is what he is doing! There are 3 equations in 5 unknowns. You can solve for 3 of the unknowns (x,y,z) in terms of the other 2 but those 2 (a and b) can be anything. To write a general formula your teacher is saying, "First,suppose a= 1, b= 0. The x,y,z= something written as a vector v1. Now, suppose a= 0, b=1. Okay, then x,y,z= something else written as a vector v2. " The general formula is then av1+ bv2. That's a linear combination that obviously gives the correct answer when a=1, b=0 and when a=0, b=1 and that's enough to show it is the general solution to this linear problem.

(Oh, the bit about letting a= 0, b= 0 is because this is not a "homogeneous" problem.)
 
  • #7
Well if that is infact what's going on, then how is this possible?>
3x+2y+z = 8;
x = a
y = b
z = 8 - 3a - 2b

Okay he let a = 0, b = 0, so z = 8
a = 1, b = 0, he has z = -3
z = 8 -3(1) - 2(0) = 5?
then he has
a = 0; b = 1; z = -2
z = 8 -3(0) -2(1) = 6?
but he has -2.
 

What are matrices and how are they used in science?

Matrices are mathematical structures that are used to organize and manipulate data in a variety of fields, including science. They are used to represent linear transformations and solve systems of equations, among other things. In science, matrices are commonly used in data analysis, computer simulations, and modeling.

How do I know when I can't cancel out any more elements in a matrix?

In order to cancel out elements in a matrix, they must be in the same position in each row. Once you have eliminated all elements in a row, you cannot cancel out any more elements in that row. You can continue to cancel out elements in other rows until you have eliminated all elements in the matrix.

Why is it important to be able to cancel out elements in a matrix?

Cancelling out elements in a matrix can simplify the matrix and make it easier to work with. It can also reveal patterns and relationships within the data that may not have been apparent before. Additionally, by cancelling out elements, you can reduce the size of the matrix, making it easier to store and process.

What are some strategies for cancelling out elements in a matrix?

One strategy for cancelling out elements in a matrix is to use elementary row operations, such as adding or subtracting rows or multiplying a row by a scalar. Another strategy is to use Gaussian elimination, which involves using a series of row operations to transform the matrix into an upper triangular form, making it easier to cancel out elements.

What are some common mistakes to avoid when trying to cancel out elements in a matrix?

One common mistake is to incorrectly perform elementary row operations, which can result in an incorrect solution. Another mistake is to overlook any elements that cannot be cancelled out, as they may still be important in the overall solution. It is also important to double check your work and make sure all operations are performed correctly.

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