Posted by **David** on Tuesday, February 9, 2010 at 5:37pm.

Suppose A is a square matrix satisfying the equation A^3 - 2I = 0. Prove that A and (A - I) respectively are invertible. (the hint is to find an explicit equation for A^-1.

To prove A is invertible, this is what I did although I don't that it's right:

A^3 - 2I = 0

A(A^2 - 2A^-1) = 0 because A * A^-1 = I

so (A^2 - 2A^-1) must = A^-1

(A^2 - 2A^-1) = A^-1

so (1/3)A^2 = A^-1 thus proving A is invertible. However I don't know that this is correct, and I don't know how to prove that (A - I) is also invertible. I know I need to factor the equation, but the 2I makes it difficult.

## Answer This Question

## Related Questions

- linear algebra - 1)If A is an invertible matrix and k is a positive integer, ...
- LINEAR ALGEBRA - How to prove or disprove (a)if A has a zeronentryonthe diagonal...
- math - If A^TA is an invertible matrix, prove that the column vectors of A are ...
- Algebra - Let A and B be n x n matrices, assume AB is invertible and show that ...
- Linear Algebra - Consider the linear transformation T: R^3->R^3 which acts by...
- math - I need help with this one... Thanks!!!! Prove the following statement: ...
- MATHS----Matrix - For a given square matrix A the predicted values of matrix B ...
- Linear Algebra - 2. Suppose that T is a linear transformation from R2 to R4 such...
- math - Suppose P is an invertible 3×3 matrix with real entries and P4=2P. Find ...
- Linear Algebra - Let T : Rn-->Rn be a linear transformation, whose standard ...

More Related Questions