Take-home midterm exam on linear algebra (2020)
� Due date/time: 5pm, Friday, Sept 18
� Submit to Canvas
1. Consider the matrix
A =
[email protected] 1 2 �10 1 1
1 0 �1
1A
by performing elementary row operations on the matrix (AjI3), …nd the inverse of A
2.
(a) Consider the following matrix
A =
[email protected]
1 1 2 �1
0 1 0 3
�1 2 �3 4
0 5 0 �2
1CCCA
i. Compute the determinant of A
ii. State whether A is invertible. Brie‡y justify your answer
(b) Let A be 3� 3 matrix and suppose that jAj = 2. Compute
i. j3Aj
ii. j3A�1j
iii. j(3A)�1j
3.
(a) Use partitioning to compute the inverse of the following matrix:
K =
[email protected]
2 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
1CCCA
(b) Let a 2 Rn with kak = 1; …nd jI + aa0j
1
4. Consider the following vectors in R3 :
v1 =
[email protected] 41
2
1A , v2 =
[email protected] 25
�5
1A , v3 =
[email protected] 2�1
3
1A
(a) Use row reduction to determine whether fv1; v2; v3g is linearly independent. If the set is not
linearly independent, give an explicit linear dependency between the vectors
(b) Let V = span fv1; v2; v3g, …nd dim(V )
(c) Let A = (v1; v2; v3), …nd rank (A)
5. Consider the following symmetric matrix:
A =
[email protected] 5 2 22 5 2
2 2 5
1A
(a) Compute the eigenvalues of A and the corresponding eigenvectors
(b) Give an orthogonal matrix H and a diagonal matrix D such that H 0AH = D
6.
(a) Is the following matrix positive semi-de…[email protected]
1 2 1 1
2 1 0 0
1 0 1 0
1 0 0 1
1CCCA
(b) Determine the value(s) of a for which the following matrix is positive de…nite, positive semi-
de…nite, negative de…nite, negative semide…nite, or inde…nite (There may be no values of a for
which the matrix satis…es some of these conditions.)
[email protected] a �1 2�1 �1 0
2 0 �4
1A
2