Maths Matrix Worksheet
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1
2Practice Midterm II
Problem 1.
For what values of p is the matrix
1 0 p
0 1 0
3 0 p
invertible?
Problem 2.
Suppose v
, . . . , v is an orthonormal basis of
and let Q be the square matrix with columns v
, . . . , v .
1
1
Calculate Q Q.
Problem 3.
Find the inverse of the matrix
0 0
1
0 1
3
.
1 2
2
Problem 4.
Is the matrix
2
p
p
p
. . . p
p
1
2
1
1
2
p
p
p
. . . p
p
2
1
2
2
. . .
2
p p
p p
. . .
p
1
2
invertible?
Problem 5.
1
0
0
0
1
7
4
Consider the subspace of
spanned by the vectors
,
,
.
2
2
5
2
2
4
(i) Find an orthonormal basis for V .
1
0
(ii) Find the projection of the vector u =
onto V .
0
1
(iii) Find V .
Problem 6.
Prove that for any subspace V of
, the complement V
is also a subspace of
.
Problem 7.
1
0
Find the matrix of the linear projection onto the subspace spanned by the vectors
0
,
2
.
1
1
Problem 8.
4
2
Assume A is a real 2
2 matrix with A
= I and det A =
1. Show that A
= I and Trace (A) = 0.
1
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