- #1
Fosheimdet
- 15
- 2
Homework Statement
Let A be a nxn matrix, and I the corresponding identity matrix, both in the real numbers ℝ. Assume that A^m=0 for a positive integer m. Show that I-A is an invertible matrix.
An invertible matrix is a square matrix that has an inverse, meaning that when multiplied by its inverse, the result is the identity matrix.
A matrix is invertible if its determinant is non-zero. This can be calculated by finding the product of the elements in the main diagonal and subtracting the product of the elements in the other diagonal.
A^m=0 means that the matrix A raised to the power of m is equal to the zero matrix, meaning that all of its elements are equal to zero.
A^m=0 is important because it allows us to use the power series expansion to show that (I-A)^{-1} exists, which then proves that I-A is invertible.
Yes, there are other methods such as calculating the rank of the matrix or using Gaussian elimination. However, using A^m=0 is a commonly used method in proving invertibility.