M310 Take Home Exam Due Date: Wed., Dec. 5 1. Let A be a 2 2 matrix dened by A = XXXXXXXXXX ; and (x; y) satisfy an equation x2 + y2 = 1. If (x0; y0) is the image of (x; y) under the matrix A, that is...

1. We are given the matrix

3 0
0 2
A
 
  
 
as well as a point  ,x y satisfying the equation 2 2 1.x y  We wish to find the equation
satisfied by the image  ', 'x y of the point  ,x y under the matrix A.

We have

' 3 0 3
.
' 0 2 2
x x x
y y y
       
        
       
Thus we have '/ 3x x and '/ 3 ,y y whence
2 2
2 2
1
' '
.
3 2
x y
x y
 
   
    
   
This is the equation of an ellipse with semi-major axis of length 3 along the x-axis and
semi-minor axis of length 2 along the y-axis. This ellipse is shown below.

2. We wish to compute  det ,nA nI where A is the n n matrix whose entries are all 1,
and nI is the n n identity matrix.
We have
1 1 1
1 1
.
1
1 1 1
n
n
n
A nI
n
 
 

  
 
 
 
Note that the sum of the entries in each row is zero. Thus we see that the rows are linearly
dependent, whence nA nI is singular and hence  det 0.nA nI 
3. We are given the matrix

2 3
2 3
2 3
2 3
1
1
.
1
1
x x x
y y y
A
z z z
w w w
 
 
 
 
 
 
We wish to show that
      det .A x y x z x w y z y w z w      

We have
2 3
2 3
2 3
2 3
1
1
det .
1
1
x x x
y y y
A
z z z
w w w

Subtracting the first row from the second, we obtain
 
2 3
2 2 3 3
2 3
2 3
2 3
2 2
2 3
2 3
1
0
det
1
1
1
0 1
.
1
1
x x x
y x y x y x
A
z z z
w w w
x x x
x y x xy y
y x
z z z
w w w
  

  
 
Subtracting the first row from the third, we obtain
 
  
2 3
2 2
2 2 3 3
2 3
2 3
2 2
2 2
2...