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MATH-4100 October 29, 2007 Exam #2 Solutions Please answer all 6 questions, showing your work in reasonable detail. Closed books; laptops
and calculators are not permitted. 1
1 Suppose the n × n matrix An has 5s along its main diagonal and 2s along the diagonal below and in its (1, n) position:
A 4 =
Find by cofactors of row 1 or otherwise the determinant of A 4 and then det An for n > 4.
By cofactors 5 ∗ 53 − 2 ∗ 23 ,similarly for n × n : 5 ∗ 5 n−^1 + (−1)n−^1 2 ∗ 2 n−^1 = 5n^ + (−1)n−^1 2 n
2 If you know that det A = 7, what is det B?
A =
row 1 row 2 row 3
,^ B^ =
row 3 + row 2 + row 1 row 2 + row 1 row 1
det
row 1 row 2 row 3
=^ −^ det
row 3 row 2 row 1
=^ −^ det
row 3 row 2 + row 1 row 1
= − det
row 3 + row 2 + row 1 row 2 + row 1 row 1
Therefore det B = − det A = − 7
4 Find the matrix of projection onto the plane x + y − z = 0. Find the projection of b=[2, 1 , 2]T onto the plane. Hint: It is helpful to select a basis in the plane.
To select a basis solve [ 1 1 − 1
]
x y z
=
one pivot; solutions
A =
AT^ A =
[ 2 − 1 − 1 2
] ,
( AT^ A
)− 1
[ 2 1 1 2
]
p =
[ 2 1 1 2
] [ 1 0 1 − 1 1 0
]
=^ P
=
An easier way to find p to project b onto the normal to the plane n = [1, 1 , −1]T^ first:
nnT^ b/nT^ n =
n
Then p = b − 13 n = [5/ 3 , 2 / 3 , 7 /3]T^.
If you start with an orthogonal basis, say
a = [1, − 1 , 0] , b = [1, 1 , 2]
then the matrix P of the projection on the plane is the sum of projections onto the basis vectors:
P = aaT^ aT^ a/(aT^ a) + bbT^ /bT^ b =
1 2
+^1 6
= 1 3
5 Given vectors
q 1 =
, a 2 =
, a 3 =
(a) Apply Gram-Schmidt to get orthonormal vectors q 1 , q 2 , and q 3
(b) After a little thinking find the projection of a 3 onto the orthogonal complement of q 1 and a 2.
Note that q 1 is already normalized; q 2 = a 2 /
2 , q˜ 3 = a 3 − (a 3 · q 1 ) q 1 − (a 3 · q 2 ) q 2 =
a 3 − ( 4 / 3 ) q 1 +
( 1 /
) q 2 =
‖q˜ 3 ‖ = 19 (2 ∗ 7
2 4 + 14
18
(b) this projection is ˜q 3 itself.
(c) The matrix below has orthogonal columns
W =
Without much computation find W −^1.
W W T^ =
,
therefore
W −^1 =
