0) is indeﬁnite because
H
2
=

8
e

1
<
0.
Thus (1
,
0) and (

1
,
0) are saddlepoints. The function has no maximum.
5. Consider the matrix
A
=
3 2 1
3 1 0
3 3 1
a
) What is the rank of
A
?
Answer:
The rank of
A
is 3. One way to see this is to compute the determinant. It is
(3)(1)(1) + (3)(3)(1) + (3)(2)(0)

(3)(1)(1)

(3)(2)(1)

(3)(3)(0) = 3 + 9

3

6 = 3. Since it
is nonzero, the matrix is invertible. As it has 3 rows, it has rank 3.
b
) How many linearly independent rows does
A
have? Columns?
Answer:
The row rank is the maximum number of linearly independent rows, the column rank
is the maximum number of linearly independent columns. Both must be equal to the rank of the
matrix. Since the rank is 3, there are 3 linearly independent rows, and 3 linearly independent
columns.
c
) How many solutions are there to the equation
A
x
=
3
2
1
?
Answer:
In part (a) we found that
A
is invertible. It follows that the system of equations has
exactly one solution. For the record, the solution is (4
/
3
,

2
,
3)
t
.
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 Spring '08
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 Economics, Linear Algebra, Derivative, Optimization, Invertible matrix, The Matrix Reloaded, Convex function

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