Checking Invertibility of A^3 Matrix - Tips & Solutions

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Discussion Overview

The discussion revolves around determining the conditions under which the matrix \( A^3 \) is invertible, specifically focusing on the values of \( k \) that affect the invertibility of the matrix \( A \). The conversation includes methods for checking invertibility, such as calculating determinants and finding ranks, and explores different approaches to the problem.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant inquires about the process for checking the invertibility of \( A^3 \) and expresses confusion over calculating its rank.
  • Another participant asserts that \( A^3 \) is invertible if and only if \( A \) is invertible, prompting a shift in focus to the conditions for \( A \).
  • A participant suggests that finding the rank of \( A \) is a simpler approach to determine its invertibility.
  • Another response challenges the necessity of finding the rank, proposing instead to use determinants to find values of \( k \) for which \( \text{det}(A) \neq 0 \), or to row reduce the augmented matrix \( [A|I] \) to Gaussian form.

Areas of Agreement / Disagreement

Participants express differing views on the methods to determine invertibility, with some advocating for the use of determinants while others suggest rank calculations or row reduction techniques. No consensus is reached on the best approach.

Contextual Notes

Participants do not clarify the specific form of matrix \( A \) or the exact values of \( k \) being considered, which may affect the discussion's conclusions.

Yankel
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Hi all,

I have the matrix A shown in the attached photo.

View attachment 3667

I need to check for which values of k, the matrix A^3 is invertible.

I tried calculating A^3, by multiplying A by itself twice. I got a nasty matrix. It makes no sense to me that now I am suppose to find it's rank, by using operations on rows. Am I missing something ? How would you solve this in the easiest way ?

Thanks !
 

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Hi Yankel,

Note $A^3$ is invertible if and only if $A$ is invertible. How would you determine the values of $k$ that make $A$ invertible?
 
Oh, I see, this was the missing part.

To find values for which A is invertible is easier. I find the rank of A, right ?
 
No, finding the rank of $A$ is unnecessary. If you can use determinants, find the values of $k$ for which $\text{det}(A) \neq 0$. If you don't know determinants, then consider row reducing the augmented matrix $[A|I]$ to Gaussian form.
 

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