Homogenous Equation: Finding 'X' with Non-Invertible Matrix A | Solution Help

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In summary, the question is asking to find 'X' in the equation AX=6,8,4 where A is a 3x3 matrix with the given values. However, A is not invertible, so the usual method of using the inverse does not work. Instead, the solution can be found by using substitution and row operations to solve for the unknown variables x, y, and z. This will result in a family of solutions for 'X'. Additionally, because A is singular, there may be multiple or no solutions.
  • #1
n0_3sc
243
1
Well here's the question:
Find 'X' where 'AX = '
6
8
4
and 'A = '
1 2 4
3 1 2
0 2 4
Now i would go ('AX')*('A^-1') which would clearly give me 'X' BUT 'A' is not invertible! So how do i do this? (by the way the ' is not part of the question).
:confused:

And then it says find the null vectors of A and hence the general solution. what do i do here?
 
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  • #2
I thought it went this way:

[tex](A^{-1})AX = (A^{-1})\left(\begin{array}{c}6\\8\\4\end\right)[/tex]

True, A is not invertible. But, if it were, that's how I would have written that.
 
Last edited:
  • #3
That is the way its meant to go, but their still has to be some way (apparently using row reduction) to solve for 'X'. becuase 'X = '
2
0
1
which works. ie
1 2 4 | 2 6
3 1 2 | 0 = 8
0 2 4 | 1 4
but how do they get 'X'? :eek:
 
  • #4
Go back and do your old fashioned solution by substitution, which is all you do with row operations. What you'll find is that you end up with an equation in two unknowns, say ax+by=c

let x=t some parameter, solve for y in terms of t, put thos back into one of the original equation to get z in terms of t - you've thus solved the equation for the family of solutions.

Other options might happen because for a singular matrix, their maybe a plane of, a line of, or no solutions.
 

What is a homogenous equation?

A homogenous equation is a mathematical equation where all the terms are of the same degree. This means that all the terms contain the same number of variables raised to the same power.

What is the purpose of solving a homogenous equation?

The purpose of solving a homogenous equation is to find the values of the variables that make the equation true. This can help in solving various problems in fields such as physics, chemistry, and engineering.

What are the methods for solving a homogenous equation?

There are various methods for solving a homogenous equation, including substitution, elimination, and matrix methods. The specific method to use depends on the complexity of the equation and the available information.

What are the steps to solve a homogenous equation?

The steps to solve a homogenous equation include identifying the variables and their degrees, rearranging the terms to group the variables, and solving for the variables using the appropriate method. The final step is to check the solution to ensure it satisfies the original equation.

How do homogenous equations differ from non-homogenous equations?

The main difference between homogenous and non-homogenous equations is that the terms in a homogenous equation all have the same degree, while in a non-homogenous equation, the terms can have different degrees. This can affect the methods used to solve the equations and the complexity of the solutions.

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