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CHEM 45/5515 – App Math fo’ P-­‐Chem I, Spring 2015 HOMEWORK 2 Due: Friday, Feb. 20, 10 am Please show all work as neatly as possible to receive credit! Do NOT use a computer program
UNLESS instructed to do so! Do not return this paper with answers on it! When turning in
MATLAB problems, provide both the script (code) and sample output (graphs and/or numerical
results) generated.
1. (6 pt)
Given the following vectors:
!
A = 1iˆ − 3 jˆ + 5kˆ
!
B = 5iˆ + 3 jˆ +1kˆ
!
C = 2iˆ − 2 jˆ + 3kˆ
!
D = −1iˆ + 2 jˆ + 3kˆ
compute the result for each of the following operations:
! ! !
a.
A+B ×C
! ! !
b.
C• A × D
! ! !
c.
B× A×D
(
)
(
(
)
)
2. (10 pt)
Consider the following three matrices:
" 2 0 −1 %
" 0 0 3 %
" −1 2 2 %
$
'
$
'
$
'
A =$ 5 1 0 '
B = $ 2 −1 2 '
C = $ 0 −2 0 '
$ 2 2 −1 '
$ 0 1 3 '
$ 4 0 0 '
#
&
#
&
#
&
WITHOUT using MATLAB:
a,
Find the inverses of the matrices using the row reduction method.
b.
Calculate the determinants of these matrices, and find their inverses using:
(A )T
A −1 = jk
det(A)
c.
Verify that these are, in fact, the inverses by multiplying each matrix by its
corresponding inverse.
3. (8 pt)
For the following set of linear equations:
4w − x + 0y + 3z = 10
−2w + 3x + y − 5z = −3
w + x − y + 2z = 2
3w + 2x − 4y + 0z = 4
a.
Reduce to row echelon form using Gaussian elimination, WITHOUT using
MATLAB.
b.
Reduce to row echelon form using MATLAB (and check your answer in part a).
HW1 – page 1
CHEM 45/5515 – App Math fo’ P-­‐Chem I, Spring 2015 HOMEWORK 2 Due: Friday, Feb. 20, 10 am 4. (6 pt)
To diagonalize a matrix, A, one can find the eigenvectors of the matrix and
construct a new matrix, X, where the columns are equal to the eigenvectors. The diagonalized
matrix, Ã, is calculated by
! = X −1AX
A
Given the following matrix, WITHOUT using MATLAB (a) determine the eigenvalues and
corresponding eigenvectors, then (b) use the eigenvectors to diagonalize the matrix.
" 8 −1 %
A =$
'
# 5 2 &
Once you have done this by hand, verify your answers using MATLAB.
5. (6 pt)
a.
b.
For two arbitrary complex numbers, z1 and z2, prove the following:
|z1 + z2| ≤ |z1| + |z2| {triangle inequality}
|z1 + z2|2 + |z1 – z2|2 = 2[ |z1|2 + |z2|2] {parallelogram equality}
HW1 – page 2