Page 332 - 35Linear Algebra

332 Sample Second Midterm

3
the vector X ∈ R . Find, but do not solve, the characteristic polynomial of
A.

3. Let M be any 2 × 2 matrix. Show

1 1
2
2
det M = − trM + (trM) .
2 2
i
4. The permanent: Let M = (M ) be an n×n matrix. An operation producing
j
a single number from M similar to the determinant is the “permanent”
X 1 2 n
perm M = M σ(1) M σ(2) · · · M σ(n) .
σ

For example

a b
perm = ad + bc .
c d
Calculate
 
1 2 3
perm  4 5 6  .
7 8 9

What do you think would happen to the permanent of an n × n matrix M
if (include a brief explanation with each answer):

(a) You multiplied M by a number λ.
(b) You multiplied a row of M by a number λ.

(c) You took the transpose of M.
(d) You swapped two rows of M.

5. Let X be an n × 1 matrix subject to

T
X X = (1) ,
and define
T
H = I − 2XX ,
(where I is the n × n identity matrix). Show

T
H = H = H −1 .

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