21.12.1. Statement

Try these first

1. 1.

   What is the Cayley-Hamilton theorem?

   Hint

   Think about matrices and their equations.

2. 2.

   Can a non-singular matrix be inverted?

   Hint

   Recall the properties of determinants.

3. 3.

   What does the Cayley-Hamilton theorem state?

   • Every matrix can be inverted
   • Every square matrix satisfies its own characteristic polynomial
   • Every matrix has a determinant

   Hint

   Think about any distinctions between types of matrices.

4. 4.

   True or False: All matrices satisfy the Cayley-Hamilton theorem.

   • True
   • False

   Hint

   Consider the definition of square matrices.

5. 5.

   Given the matrix A = [[4, 1], [2, 3]], calculate its characteristic polynomial, apply the Cayley-Hamilton theorem, and find A^{-1} with the theorem's help.

   Hint

   Utilize the determinant and characteristics from previous examples.

6. 6.

   Using the Cayley-Hamilton theorem, express A^4 in terms of A^3 and A^2 for the matrix A = [[1, 2], [0, 1]].

   Hint

   Consider the pattern formed in the powers of A to simplify.