MATH 415 / EXAM 3 test
(1) Use Cramer’s Rule to determine
x
1
.
7
5
9
7
x
1
x
2
=
1

1
.
x
1
=
1
5

1
7
/
7
5
9
7
= 12
/
4 = 3
.
The complete solution is
x
1
= 3
, x
2
=

4
,
which you can use to verify your
answer, but the answer itself should use Cramer’s Rule.
(2) Find the inverse matrix for
A
using the formula
A

1
=
C
T
/
det(
A
).
A
=
1
1
0
2
1
1
4
2
1
C
=

1
2
0

1
1
2
1

1

1
,
C
T
=

1

1
1
2
1

1
0
2

1
.
The product
AC
T
=
I
. Thus det(
A
) = 1 and
A

1
=
C
T
.
(3) Let
A
be a real symmetric matrix and let
x
and
y
be eigenvectors such that
A
x
=
λ
x
and
A
y
=
μ
y
for distinct eigenvalues
λ
and
μ
. Show that
x
and
y
are orthogonal.
y
T
A
x
=
y
T
λ
x
,
x
T
A
y
=
x
T
μ
y
.
With
A
T
=
A
the two left sides are equal. But then also the right sides are
equal which is only posible for
y
T
x
=
x
T
y
= 0
.
(4) Show that the matrix
A
is positive definite and determine its Cholesky de
composition.
A
=
1
2
1
2
8
2
1
2
5
A
=
LDL
T
=
1
0
0
2
1
0
1
0
1
1
0
0
0
4
0
0
0
4
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 Spring '08
 Staff
 Math, Linear Algebra, Algebra, Matrices, Orthogonal matrix, Cramer, distinct eigenvalues

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