Last name:
name:
1
Quizz 2 (Notes, books, and calculators are not authorized).
Show all your work in the blank space you are given on the exam sheet. Always justify your
answer. Answers with no justification will not be graded.
1 −2 3
3 0 2
Question 1: Let A =
,B=
. Find (a) A + B (b) 2A − 3B (c) AT B.
4 5 −6
−7 1 8
4 −2 5
A+B =
−3 6 2
−7 −4
0
2A − 3B =
−29 7 −36


−25 4
34
36 
AT B = −41 5
51 −6 −42
5
Question 2: Find the inverse of A =
4
3
.
2
The determinant of A is |A| = 10 − 12 = −2 6= 0. The matrix is invertible. We use the formula
from class to compute the inverse:
1
2 −3
−1
A =
−2 −4 5
2
Quizz 2, September 13, 2011
T
Question 3: Recall that a complex square matrix M is said
if MM = I. De
to be√unitary
i
−
3
1 √
1 1+i 1−i
termine which of the following matrices are unitary: A = 2
,B= 2
.
1−i 1+i
3 −i
T
We compute AA ,
T
AA =
1 i
√
3
4
√ − 3
−i
√
−i
− 3
√ 1
3
=
0
i
0
1
T
This proves that A is unitary. We compute BB , (using that (1+i)(1−i) = 2 and (1+i)2 +(1−i)2 =
0)
1 1+i 1−i 1−i 1+i
T
1 0
=
BB =
0 1
4 1−i 1+i 1+i 1−i
This proves that B is unitary.
Question 4: Find a 2×2 orthogonal matrix P whose first row is a multiple of (3, −4).
P being orthogonal means that the rows of P form an orthonormal set (see Thm 2.6, book). This
immediately implies that the second row of P is a multiple of (4, 3), since (3, −4) · (4, 3) = 0.
1 3 −4
B=
5 4 3