For sqr mtrx A w/ A^p=0 (p some natural), prove (I-A)^(-1) = I + sum[k=1,p-1] A^k

[rmp13798]

Hello

I have been taking a course in Linear Algebra (self-study) and so far I believe I have understood everything I came across (matrix operations and inverses) but this problem just slapped me in the face. It is as follows:

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Let A be a square matrix such that (A^p)=0 for some natural p. Prove the following:

. . . . .\(\displaystyle \large{ \displaystyle \left(I\, -\, A\right)^{-1}\, =\, I\, +\, \sum_{k = 1}^{p - 1}\, A^k }\)

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I tried to split the identity matrix into A and A^(-1) but that doesn't seem to work... I really have no idea how to come up with that summation...
Thanks in advance

rmp

 

[rmp13798]

Solved it

I used the inverse definition and tried it out for p=2 and p=3 to understand the logic.
I then expanded the sum into a series and was able to prove it for all values of p.
I was quite easy

 

[Prove It]

Hello

I have been taking a course in Linear Algebra (self-study) and so far I believe I have understood everything I came across (matrix operations and inverses) but this problem just slapped me in the face. It is as follows:

---

Let A be a square matrix such that (A^p)=0 for some natural p. Prove the following:

. . . . .\(\displaystyle \large{ \displaystyle \left(I\, -\, A\right)^{-1}\, =\, I\, +\, \sum_{k = 1}^{p - 1}\, A^k }\)

---

I tried to split the identity matrix into A and A^(-1) but that doesn't seem to work... I really have no idea how to come up with that summation...
Thanks in advance

rmp

Hmmm, if \(\displaystyle \displaystyle \begin{align*} A^p = 0 \end{align*}\) what does that tell you about \(\displaystyle \displaystyle \begin{align*} A \end{align*}\)?

 

[Ishuda]

Hello

I have been taking a course in Linear Algebra (self-study) and so far I believe I have understood everything I came across (matrix operations and inverses) but this problem just slapped me in the face. It is as follows:

---

Let A be a square matrix such that (A^p)=0 for some natural p. Prove the following:

. . . . .\(\displaystyle \large{ \displaystyle \left(I\, -\, A\right)^{-1}\, =\, I\, +\, \sum_{k = 1}^{p - 1}\, A^k }\)

---

I tried to split the identity matrix into A and A^(-1) but that doesn't seem to work... I really have no idea how to come up with that summation...
Thanks in advance

rmp

Suppose you were to multiply through by .\(\displaystyle \large{ \displaystyle \left(I\, -\, A\right)}\). Now back things out. What are you assuming when you do back things out?