v. This theorem provides a new method for finding inverses when they exist: substitute the matrix into the characteristic polynomial to get the equation guaranteed by your theorem, then solve for A-1 (potentially using factor- ing.) Demonstrate this technique by performing it on both nontrivial 2 x 2 and 3 x 3 matrices of your choice and comparing your inverses with those you determine by another method (such as just checking with Sage).

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