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1
Quiz 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) BAT .
4 5 6
7 1 8
4 2 5
A+B =
11 6 14
0 6 7
3A − B =
5 14 10
9 24
T
BA =
33 81
Question 2: Find the inverse of A =
2
3
4
.
7
The determinant of A is |A| = 14 − 12 = 2 6= 0. The matrix is invertible. We use the formula from
class to compute the inverse:
1 7 −4
A−1 =
2 −3 2
2
Quiz 2, September 11, 2012
T
Question 3: A complex square matrix M is said to be
if M
which
√ unitary
M = I. Determine
i
−
3
1
+
i
1
−
i
of the following matrices are unitary: A = 12 √
, B = 21
.
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 real-valued matrix P whose rows are unit orthogonal vectors and
whose first row is a multiple of (3, −4).
The first row of P is λ(3, −4) where λ ∈ R. The norm of this vector is λ2 (9 + 16) = 25λ. Since
this is a unit vector, we infer that λ = 51 . The second row of P is (a, b) with a2 + b2 = 1 and
4
3
9
4
3
3
2
5 a − 5 b = 0. This means b = 4 and ( 16 + 1)a = 1, i.e., a = 5 and b = 5 . In conclusion
1 3 −4
P =
5 4 3