Page 343 - 35Linear Algebra
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343
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(h) Is M M diagonalizable?
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(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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