Linear Algebra 2270
Ivlidterm 3
Group A
08/06/2015
Solve 4
out of
6 problems:
1. Find the LU factorization the following matrix
—1
1
2
A=
0
1
—2
1
1
1
Usiiig the factorization, find the determinant of A and its rank.
2. Find the rank and the inverse of the following matrix:
0
A= 1
2
1
1
—2
1
1
2
3. Find the determinant aiid the rank of the following matrix:
2
2
A—
2
2
—
2
4
—1
—1
3
—3
1
0
1
1
0
1
4. What is the definitioii of
(a) rank of a matrix
(b) orthonormal vectors
5. Prove that the set of
6. Matrix A
E
symmetric matrices is a subspace
IR is called symmetric
positive
of
definite (SPD) if
(a) A is symmetric
(b) for any x e R, if x *
then
Prove that if A is SPD tlieii
,
xTAx >
0
(x,y)
is an inner prodnct.
=
yTAx
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Linear Algebra 2270
Midterm 3
Group B
08/06/2015
Solve 4 out of 6 problems:
1. Find the LU factorization the following matrix
—1
1
3
A=
0
1
—2
1
1
1
Using the factorization, find the determinant of A and its rank.
2. Find the rank and the inverse of the following matrix:
011
—1 —1
2 —2 2
A= —1
3. Find the determinant and the rank of the following matrix:
A
=
2
2
2
2
3
—3
1
0
2
4
—1
—1
1
1
0
1
4. What is the definition of
(a) linearly independent vectors
(b) inverse of a matrix
5. Prove that the set of symmetric matrices is a subspace of R><
6. Consider a linear system Ax
=
b with infinitely
niany
0
Ax
=
solutions. Let xo be a solution of this system:
b
Prove that the solution set of this linear system may be written as
{ x + x : Ax =
1
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