Question

A problem or two using the Invertible Matrix Theorem, IMT. Assume all matrices are square n×n matrices. If ( A ) is invertible, then the columns of A₋¹ span Rⁿ briefly explain why using IMT

Answer 1

The columns of an invertible matrix span R^n according to the Invertible Matrix Theorem (IMT).

Step-by-step explanation:

The Invertible Matrix Theorem states that if a matrix A is invertible, then it satisfies several conditions. One of these conditions is that the columns of A span the entire space R^n. In other words, any vector in R^n can be written as a linear combination of the columns of A.

This can be explained by considering the inverse of A, denoted as A^-1. When we multiply A by A^-1, we get the identity matrix I. The columns of the identity matrix span R^n, since it has a 1 in every row and 0s elsewhere. Therefore, the columns of A^-1, which are the inverse of the columns of A, also span R^n.

Thus, if A is invertible, its columns span R^n.