Let a isinnbsp mnxn r differ from i and o if a is

Question

1. Let F be a field. Find a matrix A ∈ M_4x4(F) satisfying A4 = I ≠ A³.

2. Let n be a positive integer, let F be a field, and let A ∈ M_nxn(F) satisfy the condition A = AA^T. Show that A² = A.      

3. Let n and p be positive integers and let F be a field. Let A ∈ M_nxn(F) and let B, C ∈ M_nxp(F) be matrices satisfying the condition that A and (I + C^T A^-1 B) are nonsingular. Show that A + BC^T is nonsingular, and that

(A + BC^T)^-1 = A^-1 - A^-1B(I + C^T A^-1 B)^-1 C^T A^-1.

4. Find all solution to the system

5. Let k and n be positive integers and let F be a field. For matrices A, B ∈ Mkxn(F), show that the rank of A + B in no more than the sum of the ranks of A and of B.

6. Let n be a positive integer and let A = [a_ij] ∈  M_nxn (R) be the matrix defined by

Calculate |A|.

7. Let F be a field, let n be a positive integer, and let A = [a_ij]  ∈  M_nxn (F) be nonsingular. Show that adj(adj(A)) = |A|^n-2A.

8. Find the eigenvalues of the matrix ∈  M_3x3 (R) and, for each such eigenvalue, find the associated eigenspace.

9. Let A ∈  M_nxn (R) differ from I and O. If A is idempotent, show that its Jordan canonical form is a diagonal matrix.