Matrix manipulation (inverse, lin. alg.)

In summary, the conversation discusses a problem involving a matrix A that satisfies the equation A^3 - A^2 - I_n = 0. The conversation goes on to show that A is invertible and that the solution to another equation involving A has a unique solution. The solution involves basic algebraic manipulation and does not require advanced matrix algebra. The conversation ends with a request for help on the final steps of solving the problem.
  • #1
boings
18
0

Homework Statement



Let A[itex]\in[/itex]M[itex]_{n}[/itex]([itex]\Re[/itex]) a matrix verifying

A[itex]^{3}[/itex]-A[itex]^{2}[/itex]-I[itex]_{n}[/itex]=0

a) Show that A is inversible and calculate it
b) Show that the solution X[itex]\subset[/itex]M[itex]_{n}[/itex]([itex]\Re[/itex]) of the equation

A[itex]^{k}[/itex](A-I[itex]_{n}[/itex])X=I[itex]_{n}[/itex]

has a unique solution.





The Attempt at a Solution



I'm having trouble with starting this one. I'm quite rubbish with these matrices in linear algebra, but I have exams in a few days and this question was on it, so i need help!

I know the criteria for matrix inverse (AB=BA=I). However there's too much going on... help me dissect it? thanks a lot to anyone for any help, much appreciated.
 
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  • #2
boings said:

Homework Statement



Let A[itex]\in[/itex]M[itex]_{n}[/itex]([itex]\Re[/itex]) a matrix verifying

A[itex]^{3}[/itex]-A[itex]^{2}[/itex]-I[itex]_{n}[/itex]=0

a) Show that A is inversible and calculate it
This is kind of trivial! [itex]A^3- A^2= A(A^2- A)= (A^2- A)A= I[/itex].

b) Show that the solution X[itex]\subset[/itex]M[itex]_{n}[/itex]([itex]\Re[/itex]) of the equation

A[itex]^{k}[/itex](A-I[itex]_{n}[/itex])X=I[itex]_{n}[/itex]

has a unique solution.
Again, from [itex]A^3- A^2- I= 0[/itex], we have [itex]A^2(A- I)= I[/itex] so for any [itex]k\ge 2[/itex], [itex]A^k(I- I)X= A^{k-2}(A^2(A- I)X= A^{k- 2}X= I[/itex]. And since A has an inverse, you just multiply both sides by [itex]A^{-1}[/itex] k- 2 times. The cases where k= 0 or k= 1 are simple.

The Attempt at a Solution



I'm having trouble with starting this one. I'm quite rubbish with these matrices in linear algebra, but I have exams in a few days and this question was on it, so i need help!

I know the criteria for matrix inverse (AB=BA=I). However there's too much going on... help me dissect it? thanks a lot to anyone for any help, much appreciated.

There was really no "matrix" algebra involved here, just the definitions and basic algebraic manipulation.
 
  • #3
Oh I see, that was easier than I thought. I always get tripped up when they mention matrices and think that I have to tread very cautiously.

thank you
 
  • #4
Actually, can you help me out with the last steps for solving? I'm still a bit caught up. I attached my attempt where you left off.

thanks
 

Attachments

  • Qcontinued.jpg
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Related to Matrix manipulation (inverse, lin. alg.)

1. What is matrix manipulation?

Matrix manipulation is the process of performing operations on matrices, such as finding their inverses or solving systems of linear equations.

2. Why is finding the inverse of a matrix important?

Finding the inverse of a matrix is important because it allows us to solve systems of linear equations, which have many real-world applications in fields such as engineering, economics, and physics.

3. What is the difference between a square matrix and a non-square matrix?

A square matrix has an equal number of rows and columns, while a non-square matrix has a different number of rows and columns.

4. How do you find the inverse of a matrix?

The inverse of a matrix can be found by using the Gauss-Jordan elimination method or by finding the determinant of the matrix and applying the adjugate formula.

5. What is the purpose of using matrix manipulation in data analysis?

Matrix manipulation is used in data analysis to transform and manipulate data sets in order to extract meaningful insights and make predictions. It is particularly useful in machine learning algorithms and statistical modeling.

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