Math 107: Midterm Exam # 3
Spring 2016

−1


 5
2
4
5
Problem 1) (15 pts) A = 
 2
0
6 −3

−3 −1 −5 1





2
1
2
1a) Compute det A
1b) Determine whether column vectors of A are linearly independent.
Problem 2) (15 pts) Let A be an n × n matrix with characteristic polynomial
PA (t) = t(t − 1)2 (t + 2)3
. i) Find n.
ii) Write all distinct eigenvalues of A.
iii) What is the algebraic multiplicity of each eigenvalue of A?
iv) Find the determinant of A.
v) Suppose that A is diagonalizable. What is the geometric multiplicity of each eigenvalue of A?
Problem 3) (20 pts) Determine whether the following matrices are diagonalizable. If
diagonalizable, write the diagonalization explicitly.
"
3a) A =
1 3
#
4 2

1 2 3
4



 0 2 5 −1 

3b) B = 
 0 0 2 7 


0 0 0 3
"
Problem 4) (20 pts) Let A =
1 −2
1
#
3
4a) Write the real factorization of A.
4b) Write the diagonalization of A over C.
4c) Let B be a 5x5 matrix with characteristic polynomial λ5 − 1. Compute B50 and B51 .
Problem 5a) (12 pts) Write the diagonalization of the following symmetric matrix explicitly.


7 −4 4



A =  −4 5 0 

4
0 9
λ = 1, 7, 13
5b) (8 pts) Let A be a 4x4 symmetric matrix. Let u, v ∈ R4 such that A(u) = 3u and
A(v) = 5v. Show that u ⊥ v.
Problem 6) (15 pts) For each of (i)-(v) below: If the proposition is true, write TRUE.
If the proposition is false, write FALSE. No explanations are required for this problem.
(i) det(−A) = − det(A) for any square matrix A.
(ii) If A is invertible and 1 is an eigenvalue of A, then 1 is also an eigenvalue of A−1 .
(iii) Any eigenvector of A corresponding to a nonzero eigenvalue is in the column space
of A.
(iv) If An×n has n distinct eigenvalues λ1 , ..., λn then det(A) = λ1 .λ2 ...λn .
(v) If A is an n × n matrix with orthogonal columns, then t A = A−1 .