Can (I-A)^{-1} Be Expressed as a Series When A^4 = 0?

In summary, a polynomial problem proof is a mathematical technique used to demonstrate the truth of a polynomial equation for all possible values of the variables involved. To solve a polynomial problem proof, one must identify the equation and use techniques such as substitution or factoring to simplify it. A polynomial problem proof differs from a polynomial problem in that it is used to verify the solution to the problem. To successfully complete a polynomial problem proof, it is important to state the given equation, break the proof down into smaller steps, use mathematical principles, and double-check for accuracy. Polynomial problem proofs are commonly used in areas of science such as physics, chemistry, engineering, computer science, and economics.
  • #1
delgeezee
12
0
Let A be a square matrix,

a) show that \(\displaystyle (I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0\)

b) show that \(\displaystyle (I-A)^{-1}= I + A + A^2+...+A^n \) if \(\displaystyle
A^{n+1}= 0\)
 
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  • #2
Re: polynomial problem proof?

B is the inverse of A iff [tex] AB = BA = I [/tex]
so
try
[tex] (I -A )( I + A + A^2 + A^3) = I-A + A - A^2 + A^2 - A^3 + A^3 - A^4 = I - A^4 = I [/tex]
 

FAQ: Can (I-A)^{-1} Be Expressed as a Series When A^4 = 0?

What is a polynomial problem proof?

A polynomial problem proof is a mathematical technique used to show that a given polynomial equation or expression is true for all possible values of the variables involved. It involves using logical reasoning and mathematical principles to demonstrate the validity of the equation.

How do you solve a polynomial problem proof?

To solve a polynomial problem proof, you must first identify the polynomial equation or expression that needs to be proven. Then, you can use techniques such as substitution, factoring, or algebraic manipulation to simplify the equation and show that it holds true for all values of the variables.

What is the difference between a polynomial problem proof and a polynomial problem?

A polynomial problem proof is a method of showing that a polynomial equation is true for all possible values, while a polynomial problem is a specific equation that needs to be solved. Essentially, a polynomial problem proof is used to verify the solution to a polynomial problem.

Are there any tips for successfully completing a polynomial problem proof?

Some tips for completing a polynomial problem proof include: clearly stating the given equation or expression, breaking the proof down into smaller steps, using mathematical principles and techniques to simplify the equation, and double-checking your work for accuracy.

In what areas of science are polynomial problem proofs commonly used?

Polynomial problem proofs are commonly used in areas of science that involve mathematical modeling and analysis, such as physics, chemistry, and engineering. They can also be used in computer science and economics to solve complex equations and demonstrate the validity of mathematical models.

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