Solving a 2x2 Matrix: Can You Help?

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In summary, the conversation is about solving for x and y in a set of equations involving matrices. The conversation goes over different approaches to solving the problem and ultimately determines that there is no solution.
  • #1
wat2000
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0
solve for x and y

|x y|
|-y x|

minus

|-y x|
|x y|
equals

|4 6|
|-4 6| can someone give me a hint?
 
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  • #2
wat2000 said:
solve for x and y

|x y|
|-y x|

minus

|-y x|
|x y|
equals

|4 6|
|-4 6| can someone give me a hint?

Think about what it means for two matrices to be equal to each other.
 
  • #3
x+y=4 => y=4-x, then (4-x)-x=6
4-2x=6 => -2x=6/4=3/2 => x=-3/4
so y=4-x=4-(-3/4)=4+3/4=16/4+3/4=19/4
would this be the answer?
 
  • #4
You are on the right track with that, however be careful when you are solving the actual equations.

"4-2x=6 => -2x=6/4"

That step isn't quite correct.

Another thing that you will need to check is that the calculated values of x and y work for all 4 equations that you could write. Have you missed out a minus sign by any chance on one of the 6's? The reason i ask is because we can write the following two equations:

y - x = 6
x - y = 6

No value of x and y will work with these
 
  • #5
I tried something different and got x=-1 and y= 5 is that right.
 
  • #6
That's not what I get. You should get four equations in two unknowns.
Can you confirm that what you wrote is exactly what the problem is?
 
  • #7
wat2000 said:
I tried something different and got x=-1 and y= 5 is that right.

Yes they would solve the first two equations, however they still do not solve the second two. You need all elements of the matrix to be consistent, so you need solutions to all four equations:

x+y = 4
y-x = 6

You solved those two, but the solution needs to also solve the remaining two:

-y-x = -4
x-y = 6

If you substitute x=-1 and y=5 into the top equation then you get -4 which is ok, but the second equation gives -1-5 = -6 which is inconsistent. Either there is no solution to the whole problem or you have mis-typed the 6 instead of a -6 perhaps?
 
  • #8
x-(-y)=4
y-x=6
-y-x=-4
x-y=6

4-2x=6
-2x=6-4
-2x=2
x=-1

since y=4-x, then y=4-(-1)
4+1=5
y=5
check: |-1 -5|
|-5-1| minus

|-5 -1|
|-1 5| equals

|4 6|
|-4 6|
Thats how I got it.
 
  • #9
[tex]

\left[ {\begin{array}{*{20}c}
{ - 1} & 5 \\
{ - 5} & { - 1} \\
\end{array}} \right] - \left[ {\begin{array}{*{20}c}
{ - 5} & { - 1} \\
{ - 1} & 5 \\
\end{array}} \right] = \left[ {\begin{array}{*{20}c}
4 & 6 \\
{ - 4} & { - 6} \\
\end{array}} \right] \ne \left[ {\begin{array}{*{20}c}
4 & 6 \\
{ - 4} & 6 \\
\end{array}} \right][/tex]
 
  • #10
Where did I mess up?
 
  • #11
Is the question you wrote definitely correct?
 
  • #12
yes. is there just no solution?
 
  • #13
None that i can see.
 
  • #14
ok that's what I thought. Thanks for your help.
 

FAQ: Solving a 2x2 Matrix: Can You Help?

1. How do you solve a 2x2 matrix?

To solve a 2x2 matrix, you will need to use the method of elimination or substitution. First, rearrange the equations so that the variables are on one side and the constants are on the other. Then, solve for one variable and plug that value into the other equation to solve for the second variable. Finally, check your solution by plugging it back into both equations to ensure it satisfies both equations.

2. What is the purpose of solving a 2x2 matrix?

Solving a 2x2 matrix allows you to find the values of the variables in a system of linear equations. This can be useful in many real-world applications, such as finding the intersection point of two lines or determining the break-even point in a business scenario.

3. Can a 2x2 matrix have more than one solution?

Yes, a 2x2 matrix can have infinitely many solutions, one solution, or no solution at all. This depends on the coefficients and constants in the equations. If the equations are consistent and independent, there will be one unique solution. If the equations are consistent but dependent, there will be infinitely many solutions. If the equations are inconsistent, there will be no solution.

4. What is the difference between a 2x2 matrix and a 2x2 determinant?

A 2x2 matrix is a rectangular array of numbers with two rows and two columns, while a 2x2 determinant is a single number calculated from the elements of a 2x2 matrix. The determinant is found by multiplying the two diagonal elements and subtracting the product of the two other elements.

5. Is there a specific order in which to solve a 2x2 matrix?

There is no specific order in which to solve a 2x2 matrix, as long as you use a consistent method, such as elimination or substitution. However, it is generally easier to solve for the variable with the smallest coefficient first, as it will result in simpler calculations. Additionally, if one equation has a coefficient of 1 for one of the variables, it may be easier to solve for that variable first.

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