Question

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 �1 0 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 =

0BBB@ 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 =

0BBB@ 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 =

0@ 4

1

2

1A , v2 = 0@ 2

5

�5

1A , v3 = 0@ 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 =

0@ 5 2 2

2 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�nite?0BBB@ 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.)

0@ a �1 2

�1 �1 0

2 0 �4

1A

2