If AB^2-A Invertible Prove that BA-A Invertible

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In summary, AB^2-A Invertible refers to a matrix property where it can be multiplied by another matrix to produce the identity matrix. Similarly, to prove that BA-A Invertible means to show that the matrix BA-A has an inverse matrix that can be multiplied by it to give the identity matrix. To prove that a matrix is invertible, one can show that its determinant is non-zero or find its inverse using methods like row reduction or the adjugate matrix. The importance of proving a matrix is invertible lies in its ability to solve systems of linear equations and perform operations on matrices. If a matrix is not invertible, it cannot be used for these purposes and may represent a singular or degenerate system.
  • #1
ThankYou
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Homework Statement


As in the tile
If AB^2-A is a Invertible matrix Prove that BA-A is also a Invertible matrix



Homework Equations


liner algebra 1 , only the start...


The Attempt at a Solution


Made many things noting work

Thank you
 
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  • #2
I've solve it..
Thank you..
AB^2-A = A(B^2-I)
Means that A and (B^2-I) are Invertible
(B-I)(B+I) are Invertible and so
A(B-I) Is Invertible
 
  • #3
ThankYou said:
I've solve it..
Thank you..
AB^2-A = A(B^2-I)
Means that A and (B^2-I) are Invertible
(B-I)(B+I) are Invertible and so
A(B-I) Is Invertible
You're supposed to show that BA - A is invertible.

A(B - I) = AB - A, which is not necessarily equal to BA - A.
 
  • #4
Nevertheless, we can save this reasoning provided you are allowed to use det(AB)=det(A)det(B).
 
  • #5
I am allowed
 
  • #6
Then we can just say det(AB2-A)=det(A)det(B-I)det(B+I) is nonzero, so det(BA-A)=det(B-I)det(A) is nonzero.
 

What is the meaning of AB^2-A Invertible?

AB^2-A Invertible refers to the property of a matrix where it can be multiplied by another matrix to produce the identity matrix. This means that the matrix is "invertible" or has an inverse matrix that can be multiplied by it to give the identity matrix.

What does it mean to prove that BA-A Invertible?

To prove that BA-A Invertible means to show that the matrix BA-A is invertible, or has an inverse matrix that can be multiplied by it to give the identity matrix. This is usually done by showing that the determinant of the matrix is non-zero.

How can I prove that a matrix is invertible?

To prove that a matrix is invertible, you can show that the determinant of the matrix is non-zero. This means that the matrix can be multiplied by another matrix to give the identity matrix. Another way is to find the inverse of the matrix by using various methods such as row reduction or using the adjugate matrix.

What is the importance of proving that a matrix is invertible?

Proving that a matrix is invertible is important because it allows us to find the inverse of the matrix and solve systems of linear equations. It also helps in performing various operations on matrices, such as finding the determinant and eigenvalues.

What happens if a matrix is not invertible?

If a matrix is not invertible, it means that it does not have an inverse matrix and therefore, cannot be used to solve systems of linear equations. It also means that certain operations, such as division, cannot be performed on the matrix. In some cases, a non-invertible matrix may represent a singular or degenerate system, which has no unique solution.

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