Reference no: EM131131470
1. Let F be a field. Find a matrix A ∈ M4x4(F) satisfying A4 = I ≠ A3.
2. Let n be a positive integer, let F be a field, and let A ∈ Mnxn(F) satisfy the condition A = AAT. Show that A2 = A.
3. Let n and p be positive integers and let F be a field. Let A ∈ Mnxn(F) and let B, C ∈ Mnxp(F) be matrices satisfying the condition that A and (I + CT A-1 B) are nonsingular. Show that A + BCT is nonsingular, and that
(A + BCT)-1 = A-1 - A-1B(I + CT A-1 B)-1 CT 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 = [aij] ∈ Mnxn (R) be the matrix defined by
Calculate |A|.
7. Let F be a field, let n be a positive integer, and let A = [aij] ∈ Mnxn (F) be nonsingular. Show that adj(adj(A)) = |A|n-2A.
8. Find the eigenvalues of the matrix 
∈ M3x3 (R) and, for each such eigenvalue, find the associated eigenspace.
9. Let A ∈ Mnxn (R) differ from I and O. If A is idempotent, show that its Jordan canonical form is a diagonal matrix.
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