Get Expert Help with Invertible Matrices: Tips and Techniques

[FancyChancey]

Hi all. I'm having a real tough time with this question. I don't know where to begin and how to go about the question. Can someone point me in the right direction please?

(Problem is on the attachment)

 

[HallsofIvy]

haven't we seen this before?

a) If A is invertible, then (A+ 2A^-1) is invertible.
When does there exist a matrix, B, such that (A+ 2A^-1)B= I?
If I remember correctly, a matrix is invertible if and only if it's determinant is not 0. What is the determinant of A+ 2A^-1?

b) If a real matrix B satisfies B²- 2B+ 1= 0, then B-I cannot be invertible.
B²- 2B+I= (B- I)²

c) There is a non-invertible matrix, B, satisfying B²- 2B+ I= 0.
As I said, B²-2B+ I= (B-I)². Therefore B= ? or ?. Is either of those non-invertible?

 

[Architeuthis Dux]

Hi there,

I understand that you are struggling with understanding how to approach a question involving invertible matrices. First of all, don't worry, it is a common problem and there are definitely ways to help you out.

To begin with, let's define what an invertible matrix is. An invertible matrix is a square matrix (meaning it has the same number of rows and columns) that can be inverted, or turned into an identity matrix when multiplied by its inverse. In simpler terms, an invertible matrix is one that has a unique solution for its inverse.

Now, for your question, the first step would be to check if the given matrix is indeed invertible. This can be done by calculating its determinant, which should not be zero for an invertible matrix.

Next, you can use various techniques to find the inverse of the matrix, such as using the adjugate matrix method or the Gaussian elimination method. These techniques involve manipulating the given matrix through certain operations to arrive at its inverse.

I would also suggest seeking help from a tutor or a classmate who is familiar with invertible matrices. They can provide you with step-by-step guidance and clarify any doubts you may have.

Remember, practice makes perfect, so don't be discouraged if it takes some time to fully understand invertible matrices. Keep practicing and seeking help, and you will eventually master this topic.

I hope this helps point you in the right direction. Good luck!