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Find the general solutions of the systems whose augmented matrices are given in Exercises $7-14$ .$\left[\begin{array}{cccccc}{1} & {2} & {-5} & {-6} & {0} & {-5} \\ {0} & {1} & {-6} & {-3} & {0} & {2} \\ {0} & {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right]$

$\left\{\begin{array}{c}{x_{1}=-7 x_{3}-9} \\ {x_{2}=6 x_{3}+3 x_{4}+2} \\ {x_{3} \text { is free }} \\ {x_{4} \text { is free }} \\ {x_{5}=0}\end{array}\right.$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Poorva N.

March 10, 2022

the question and the explanation is different

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this question is asking us to determine whether the homogenous system given tow us has a nontrivial solution and they want us to do that without doing any math out. So what's important is to know the difference between trivial and non tribunal so trivial is basically a solution with all zeros. So what that means is that given any variables x one x two x tree all the way to extend all of these values when we solve our system of equations are gonna be with a serious well of X one equals zero extreme equals zero all the way. Thio accent equal zero nontrivial solution means that one of these values so either x one next to all the way to extend one or more of these values is not equal to zero. So now that we know the definition, we can take a look at our Russian. So the first thing to know is that the last question. So let's look at the last equation that only involves the extra variable. So here we have a number. We can call that number, eh? Times x three is equal to zero. The solving for X three. We get extra is equal to zero divided by a and zero divided by any numbers in vehicle A zero. So that means that X three is equal to zero. We can now do this for X two and solve for X, too. So we have some. We have some number times x two minus some number times x three is equal to zero. But since we already know that X three is equal to zero, we know that this equal zero, which means that in order for this to be true X to house to also equal zero. So now we go ahead and look at X one. So we have some number times x one plus some number Tom text two plus some number times x three is equal to zero. We know that X three is equal to zero. The next to is equal to zero said these both cancel out. You're both equal to zero and we're left with X one equals zero, which means that X one has to be equals zero. So that means that our solution for X one x you and X three is just equal to the zero vector. So every single very blows equal to zero. And if we check our definition from before of trivial versus non attributable, we can see that all of these variables how swollen extremities three. Our bull are their own zero solution. So that means that the homogeneous system that was given to us on Lee has trivial solutions.

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