-la.1.1.19 the augmented process

• MHB
• karush
In summary, the conversation is about solving two problems (19 and 21) using augmented matrices. The first problem requires finding the value of $h$ that satisfies the augmented matrix, while the second problem involves adding two rows to find the value of $h$ and solving for $y$. The conversation also touches on the concepts of linear consistency and row echelon form. Ultimately, the value of $h$ is determined to be 6 for both problems.
karush
Gold Member
MHB
nmh{1000} ov 346.21

ok I am trying to these 4 problems
but don't think I understand the augmented process

these are the answers to 19 and 21

$\tiny{1.1.19}$
$$A_{19}=\left[\begin{array}{rrrrr} 1& \,h& \,4\\ 3& \, 6& \, 8 \end{array}\right]$$$\textit{so if we multiply$R_1$by subtract from R_3 we have}$\\
$$\left[\begin{array}{rrrrr} 1& \,h& \,4\\ 0& \, (6-3h)& \, -4 \end{array}\right]$$uhmmm ?

Last edited:
karush said:
$\tiny{1.1.19}$
$$A_{19}=\left[\begin{array}{rrrrr} 1& \,h& \,4\\ 3& \, 6& \, 8 \end{array}\right]$$$\textit{so if we multiply$R_1$by subtract from R_3 we have}$
$$\left[\begin{array}{rrrrr} 1& \,h& \,4\\ 0& \, (6-3h)& \, -4 \end{array}\right]$$uhmmm ?

Good. Can you continue?
What would $y$ become?

I like Serena said:
Good. Can you continue?
What would $y$ become?

with (h=y)
\begin{align*}
(6-3h)&=-4 \\
3h&=10\\
h&=\frac{10}{3}
\end{align*}but how does this answer $h \ne 2$ ?

karush said:
with (h=y)
\begin{align*}
(6-3h)&=-4 \\
3h&=10\\
h&=\frac{10}{3}
\end{align*}

but how does this answer $h \ne 2$ ?

The second line in the augmented matrix actually stands for:
$$0\cdot x + (6-3h)\cdot y = -4 \quad\Rightarrow\quad (6-3h)y=-4$$
Can we solve $y$ from it?

Re: la.1.1.19 the augumented process

I like Serena said:
The second line in the augmented matrix actually stands for:
$$0\cdot x + (6-3h)\cdot y = -4 \quad\Rightarrow\quad (6-3h)y=-4$$
Can we solve $y$ from it?
\begin{align*} 0\cdot x + (6-3h)\cdot y &= -4\\ (6-3h)\cdot y&=-4\\ y&=-\frac{4}{6-3h} \end{align*}
so then h equals all real numbers except 2
whatever the notation is for that?

Re: la.1.1.19 the augumented process

karush said:
\begin{align*} 0\cdot x + (6-3h)\cdot y &= -4\\ (6-3h)\cdot y&=-4\\ y&=-\frac{4}{6-3h} \end{align*}
so then h equals all real numbers except 2
whatever the notation is for that?

Good. It means that $y$ has a solution if and only if $h\ne 2$.

Btw, for the system to be linearly consistent, we also need to be able to find a solution for $x$.
Can we?

Re: la.1.1.19 the augumented process

I like Serena said:
Good. It means that $y$ has a solution if and only if $h\ne 2$.

Btw, for the system to be linearly consistent, we also need to be able to find a solution for $x$.
Can we?

I can only see that x could be any real number since it would be gone by multiplying by zero

ok here is the last one but only see a single answer?

$A_{22}=\left[\begin{array}{rrrrr} -4& \,12& \,\textbf{h}\\ 2& -6& -3 \end{array}\right]\\$
$\text{Add$R_2$to$R_1$}\\$
$\left[\begin{array}{rrrrr} -4+2& \,12-6& \, h-3\\ 2& -6& -3\end{array}\right]\\ 0+0=h-3\\$
$\text{so$h=3$}$

Re: la.1.1.19 the augumented process

karush said:
I can only see that x could be any real number since it would be gone by multiplying by zero

What about the first line of the augmented matrix?
Can't we deduce $x$ from it now that we have an expression for $y$?

karush said:
ok here is the last one but only see a single answer?

$A_{22}=\left[\begin{array}{rrrrr} -4& \,12& \,\textbf{h}\\ 2& -6& -3 \end{array}\right]\\$
$\text{Add$R_2$to$R_1$}\\$
$\left[\begin{array}{rrrrr} -4+2& \,12-6& \, h-3\\ 2& -6& -3\end{array}\right]\\ 0+0=h-3\\$
$\text{so$h=3$}$

How about bringing it into row echelon form as before?
And solve for $y$ again?

Re: la.1.1.19 the augumented process

How about bringing it into row echelon form as before? And solve for $y$ again?[/QUOTE said:
$\left[\begin{array}{rrrrr} -4& \,12& \,\textbf{h}\\ 2& -6& -3 \end{array}\right]\\$
Ok to avoid fraction switch $R_1$ and $R_2$
$\left[\begin{array}{rrrrr} 2& -6& -3 \\ -4& \,12& \,\textbf{h} \end{array}\right]\\$
then multiply R_1 by 2 and add it to R_2
$\left[\begin{array}{rrrrr} 2& -6& -3 \\ 0& \,0 & \,\textbf{h-6} \end{array}\right]\\$
then
0=h-6
so
h=6

Re: la.1.1.19 the augumented process

karush said:
$\left[\begin{array}{rrrrr} -4& \,12& \,\textbf{h}\\ 2& -6& -3 \end{array}\right]\\$
Ok to avoid fraction switch $R_1$ and $R_2$
$\left[\begin{array}{rrrrr} 2& -6& -3 \\ -4& \,12& \,\textbf{h} \end{array}\right]\\$
then multiply R_1 by 2 and add it to R_2
$\left[\begin{array}{rrrrr} 2& -6& -3 \\ 0& \,0 & \,\textbf{h-6} \end{array}\right]\\$
then
0=h-6
so
h=6

Correct.
In this case we don't really get to the point that we can solve for $y$.
We can already see that $h$ must be $6$ because otherwise there can't be a solution.

https://dl.orangedox.com/wlKD7eKSWiQ79alYD6

1. What is the augmented process in scientific research?

The augmented process, also known as augmented research, is a method of scientific inquiry that involves the integration of artificial intelligence and machine learning techniques to supplement and enhance the traditional scientific process. It allows for the extraction of meaningful insights and patterns from large and complex datasets that would be difficult or impossible to identify using traditional methods.

2. How does the augmented process differ from traditional scientific methods?

The augmented process differs from traditional scientific methods in that it utilizes advanced technologies such as artificial intelligence, machine learning, and big data analytics to analyze and interpret data. It also allows for a more iterative and dynamic approach to research, where hypotheses can be continuously refined and tested as new data becomes available.

3. What are the benefits of using the augmented process in scientific research?

The augmented process has several benefits, including the ability to process and analyze large amounts of data quickly and accurately, identify patterns and connections that may not be apparent to humans, and generate new hypotheses for further exploration. It also allows for more efficient and cost-effective research, as well as the potential for new discoveries and innovations.

4. Are there any limitations or challenges to using the augmented process?

While the augmented process has many potential benefits, there are also some limitations and challenges to consider. These may include the need for large and high-quality datasets, potential biases in the data or algorithms used, and the potential for overreliance on technology rather than human intuition and expertise. Additionally, there may be ethical concerns surrounding the use of artificial intelligence and machine learning in research.

5. How is the augmented process being used in different fields of scientific research?

The augmented process is being used in a variety of fields, including biology, chemistry, physics, neuroscience, and social sciences. In biology, for example, it is being used to analyze and interpret genetic data, while in neuroscience, it is being used to understand brain function and behavior. It is also being applied in drug discovery, climate science, and other areas of research where large and complex datasets need to be analyzed and interpreted.

Replies
7
Views
1K
Replies
5
Views
1K
Replies
2
Views
810
Replies
3
Views
943
Replies
4
Views
2K
Replies
5
Views
1K
Replies
2
Views
1K
Replies
2
Views
956
Replies
14
Views
1K
Replies
2
Views
2K