Page 343 - 35Linear Algebra

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(h) Is M M diagonalizable?
T
(i) Does M M have a zero eigenvalue?
(j) Suppose U = V and ker L 6= {0 U }. Find an eigenvalue of M.
(k) Suppose U = V and ker L 6= {0 U }. Find det M.

4. Consider the system of equations

x + y + z + w = 1
x + 2y + 2z + 2w = 1
x + 2y + 3z + 3w = 1

Express this system as a matrix equation MX = V and then find the solution
set by computing an LU decomposition for the matrix M (be sure to use
back and forward substitution).

5. Compute the following determinants

 
1 2 3 4
 
1 2 3

1 2  5 6 7 8 
det , det  4 5 6  , det   ,
3 4  9 10 11 12 
7 8 9
13 14 15 16
 
1 2 3 4 5
6 7 8 9 10
 
 
det  11 12 13 14 15  .
 
16 17 18 19 20
 
21 22 23 24 25
Now test your skills on
 
1 2 3 · · · n
 n + 1 n + 2 n + 3 · · · 2n 
 

 2n + 1 2n + 2 2n + 3 3n  .
det 
.
. .
 . . 
 . . . . 
2
2
2
n − n + 1 n − n + 2 n − n + 3 · · · n 2
Make sure to jot down a few brief notes explaining any clever tricks you use.
6. For which values of a does
  
  
 1 1 a 
3
1
,
0
U = span    2    = R ?
,
1 −3 0
 
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