Math(linear algebra)
Take-home midterm exam on linear algebra (2020)
� Due date/time: 5pm, Friday, Sept 18
� Submit to Canvas
1. Consider the matrix
A =
0 @ 1 2 �10 1 1 1 0 �1
1 A
by performing elementary row operations on the matrix (AjI3), �nd the inverse of A
2.
(a) Consider the following matrix
A =
0 BBB@
1 1 2 �1 0 1 0 3
�1 2 �3 4 0 5 0 �2
1 CCCA
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 =
0 BBB@ 2 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
1 CCCA
(b) Let a 2 Rn with kak = 1; �nd jI +aa0j
1
4. Consider the following vectors in R3 :
v1 =
0 @ 41 2
1 A , v2 =
0 @ 25 �5
1 A , v3 =
0 @ 2�1
3
1 A
(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 = spanfv1;v2;v3g, �nd dim(V )
(c) Let A = (v1;v2;v3), �nd rank (A)
5. Consider the following symmetric matrix:
A =
0 @ 5 2 22 5 2 2 2 5
1 A
(a) Compute the eigenvalues of A and the corresponding eigenvectors
(b) Give an orthogonal matrix H and a diagonal matrix D such that H0AH = D
6.
(a) Is the following matrix positive semi-de�nite?
0 BBB@ 1 2 1 1
2 1 0 0
1 0 1 0
1 0 0 1
1 CCCA
(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.)
0 @ a �1 2�1 �1 0
2 0 �4
1 A
2