Suppose that a sequence satisfies the recurrence formula,
where and are real numbers, and is an integer greater than or equal to 4. Let . Answer the following questions, where is an integer greater than or equal to 4. is the identity matrix of order 3, and stands for the transpose of a matrix .
(1) Find a matrix which satisfies .
(2) Find and for which is invertible. For the values and , determine .
(3) Let and . Compute the eigenvalues of , and .
(4) With the same condition as (3), express using , and compute using the results of (1) – (3).
In the following, we assume that all elements of matrices are real numbers. In what follows, a matrix with rows and columns is called an matrix. Let be the zero matrix of rows and columns, and be the identity matrix of order . The rank of a matrix is denoted by . Let be a matrix, and be an matrix. Prove each of the following statements (1) to (5) if it holds, or if it does not hold, give a counterexample of a pair and disprove the statement with it.