Find the determinants of the following matrices

*** (PARTIAL WORK IS
NOT ACCEPTED) ***

1)
Find the determinants of the following matrices:
a)
a rank one matrix A
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b)
the upper triangular matrix
.0/msohtmlclip1/01/clip_image004.png”>
c)
The lower triangular matrix, UT.
d)
The inverse matrix, U-1.

e)
The reverse triangular matrix.
.0/msohtmlclip1/01/clip_image006.png”>
f)
Use the cofactor formula to find the inverse of matrix B.
.0/msohtmlclip1/01/clip_image008.png”>
g)
use the big formula to find the determinant of matrix C (don’t include 0 terms)
.0/msohtmlclip1/01/clip_image010.png”>

2)
Given a line in R3
that passes through the points (x0,y0,z0)
and (x’,y’,z’), and a plane in R3 defined by the three
points that are not collinear, (x1,y1,z1),
(x2,y2,z2),
(x3,y3,z3),
use Cramer’s rule to find the intersection, (x,y,z), of the line and the plane. Hint: Use the following
parametric equations for the line & plane, where t, u and v are unknown.
.0/msohtmlclip1/01/clip_image012.png”> .0/msohtmlclip1/01/clip_image014.png”>

3)
Determine whether each of these matrices [A] and [B] are:
Invertible,
Orthogonal, Projection, Symmetric, Positive Definite, Stochastic
.0/msohtmlclip1/01/clip_image016.png”> .0/msohtmlclip1/01/clip_image018.png”>

4)
Find the orthogonal matrix that diagonalizes [A].
.0/msohtmlclip1/01/clip_image020.png”>

5) The pattern of sunny and rainy
days on the island of Markov is an ever-repeated sequence with two states. Every
sunny day is followed by another sunny day with probability of 0.8. Every rainy
day is followed by another rainy day with a probability of 0.6. Today is sunny
on Markov. What is the chance of rain the day after tomorrow? Also, compute the
probability that April 1 of next year is rainy on Markov.

6) For the system of inhomogeneous
differential equations.
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and the initial condition, .0/msohtmlclip1/01/clip_image024.png”> and .0/msohtmlclip1/01/clip_image026.png”>:
a)
Arrange the system into matrix form.
b)
Find the diagonal or Jordan form of the system matrix.
c)
Write the general solution in the form of the matrix
exponential.
d)
Use the initial condition to find the solution .0/msohtmlclip1/01/clip_image028.png”> and .0/msohtmlclip1/01/clip_image030.png”>.
Receive extra credit if you do the
integral for the inhomogeneous term.

7) Consider the following electrical circuit shown in the
figure below. The circuit contains 7 resistors with resistances, R1– R7, one
voltage source, V0, and
one current source, I0.
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a)
Set up the incidence matrix for this network.

b)
Describe the subspaces of this matrix in terms of
Kirchhoff’s voltage law, Kirchhoff’s

c)
Current law, Ohm’s law, and current loops.

d)
Set up the complete system with applied loads (voltage
& current sources).

e)
Use MATLAB to solve the system. Let R1=R2=R7=
3W, R3=2W, and R4=R5=R6=1
W, V0=10V, and I0=1A.

Useful Integration
Formulas

.0/msohtmlclip1/01/clip_image034.png” alt=”Integrals #12″>

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