Math(linear algebra)
How to solve these problems and detailed steps to solve them.
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
[emailprotected]
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. Briey justify your answer
(b) Let A be 33 matrix and suppose that jAj = 2. Compute
i. j3Aj
ii. j3A1j
iii. j(3A)1j
3.
(a) Use partitioning to compute the inverse of the following matrix:
K =
0
[emailprotected]
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
@ 21
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-denite?
0
[emailprotected]
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 denite, positive semi-
denite, negative denite, negative semidenite, or indenite (There may be no values of a for
which the matrix satises some of these conditions.)
0
@ a 1 21 1 0
2 0 4
1
A
2
