A question in prooving invertion

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In summary: P\left(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\right)P?In summary, the conversation discusses proving the invertibility of an operator and finding its inverse using a standard basis. It also mentions a possible simpler method and explains the concept of an operator being its own inverse. It is concluded that the given result is correct and logical.
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transgalactic
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i need to proove that this operator is invertable
i solved it by putting a standart basis and findind that each line in the matrix
is independant.

is it ok??

is there any easier way to slve it??

when i tried to find the invert of this operator i got the same matrix as before
very weird result
is that ok??
 
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  • #2
[tex]P\left(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\right)= \left[\begin{array}{cc}d & c \\ b & a\end{array}\right][/tex]
when i tried to find the invert of this operator i got the same matrix as before
very weird result
is that ok??
Yes, that's okay. The multiplicative inverse of 1 is 1 and the multiplicative inverse of -1 is -1. The inverse of the identity operator is obviously the indentity operator. It's quite possible for for a linear operator to be its own inverse.

And it isn't necessary to write this in terms of a basis. You can "reason" it out:

We can think of P as doing two things to a matrix: 1) Swap the columns so that
[tex]\left[\begin{array}{cc}a & b \\c & d\end{array}\right][/tex]
becomes
[tex]\left[\begin{array}{cc}b & a \\d & c\end{array}\right][/tex]
Then Swap rows so that becomes
[tex]\left[\begin{array}{cc}d & c \\b & a\end{array}\right][/tex]

The inverse must do just the opposite, going back from
[tex]\left[\begin{array}{cc}d & c \\b & a\end{array}\right][/tex]
to
[tex]\left[\begin{array}{cc}a & b \\c & d\end{array}\right][/tex]

But the inverse (opposite) of "swap the rows" is "swap the rows" and the inverse of "swap the columns" is "swap the columns"!
What is
[tex]P\left(P\left(\left[\begin{array}{cc}a & b \\c & d\end{array}\right]\right)\right)[/tex]
 

Related to A question in prooving invertion

1. What is inversion in scientific terms?

Inversion, in scientific terms, refers to the process of reversing the order of a sequence of elements or molecules. This can occur in a variety of scientific fields, such as genetics, chemistry, and physics.

2. Why is proving inversion important in scientific research?

Proving inversion is important in scientific research because it helps to better understand the properties and behavior of elements and molecules. It also allows scientists to make predictions and develop new theories based on the knowledge gained from proving inversion.

3. How do scientists prove inversion?

Scientists use a variety of techniques to prove inversion, depending on the specific field of study. In genetics, for example, they may use techniques such as genetic crosses and DNA sequencing. In chemistry, they may use spectroscopy or chromatography. In physics, they may use mathematical equations and experiments.

4. What are some real-world applications of proving inversion?

Proving inversion has many real-world applications, such as developing new medicines, creating more efficient and sustainable energy sources, and understanding the origins of the universe. It is also used in everyday technology, such as in computer programming and telecommunications.

5. Are there any limitations or challenges in proving inversion?

Yes, there can be limitations and challenges in proving inversion. Some elements or molecules may be difficult to manipulate or observe, and there may be unexpected results or errors in the experiments. Additionally, proving inversion may require advanced technology or specialized knowledge, which may not be easily accessible to all scientists.

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