Algebra liniowa z geometrią
Studia internetowe
Zadania domowe #6
1. Obliczyć wyznacznik macierzy metodą permutacyjną:
A
=
[
0
0
0
2
0
0
−1 0
0
3
0
0
−2 0
0
0
]
1 2 3 4
+
a
11
a
22
a
33
a
44
0
1 2 4 3
-
a
11
a
22
a
34
a
43
0
1 3 2 4
-
a
11
a
23
a
32
a
44
0
1 3 4 2
+
a
11
a
23
a
34
a
42
0
1 4 2 3
+
a
11
a
24
a
32
a
43
0
1 4 3 2
-
a
11
a
24
a
33
a
42
0
2 1 3 4
-
a
12
a
21
a
33
a
44
0
2 1 4 3
+
a
12
a
21
a
34
a
43
0
2 3 1 4
+
a
12
a
23
a
31
a
44
0
2 3 4 1
-
0
2 4 1 3
-
0
2 4 3 1
+
0
3 1 2 4
+
0
3 1 4 2
-
0
3 2 1 4
-
0
3 2 4 1
+
0
3 4 1 2
+
0
3 4 2 1
-
0
4 1 2 3
-
0
4 1 3 2
+
0
4 2 1 3
+
0
4 2 3 1
-
0
4 3 1 2
-
0
4 3 2 1
+
12
∣
A
∣
=12
2. Obliczyć z definicji wyznacznik macierzy:
a)
A
=
[
5 0 1
0 2 0
3 0 4
]
∣A∣=−1
1
1
⋅5⋅
[
2 0
0 4
]
0⋅−1
1
3
⋅1⋅
[
0 2
3 0
]
=5⋅81⋅−6=34
b)
A
=
[
2 0
0
0
0 0
−1 0
0 3
0
0
0 0
0
2
]
∣A∣=−1
1
1
⋅2⋅
[
0
−1 0
3
0
0
0
0
2
]
=2⋅−1
1
2
⋅−1⋅
[
3 0
0 2
]
=2⋅6=12
3. Sprawdzić czy macierz A jest nieosobliwa i ewentualnie wyznaczyć macierz odwrotną
A
−1
A
=
[
2
2
−1
−1
0
1
5
−1 −6
]
∣A∣=−1⇒ macierz jest nieosobliwa
D
11
=−1
1
1
⋅
[
0
1
−1 −6
]
=1
D
12
=−1
1
2
⋅
[
−1
1
5
−6
]
=−1
D
13
=−1
1
3
⋅
[
−1
0
5
−1
]
=1
D
21
=−1
2
1
⋅
[
2
−1
−1 −6
]
=13
D
22
=−1
2
2
⋅
[
2
−1
5
−6
]
=−7
D
23
=−1
2
3
⋅
[
2
2
5
−6
]
=12
D
31
=−1
3
1
⋅
[
2
−1
0
1
]
=2
D
32
=−1
3
2
⋅
[
2
−1
−1 1
]
=−1
D
33
=−1
3
3
⋅
[
2
2
−1 0
]
=2
A
D
=
[
1
−1 1
13
−7 12
2
−1 2
]
T
=
[
1
13
2
−1 −7 −1
1
12
2
]
A
−1
=
A
D
∣A∣
=
[
−1 −13 −2
1
7
1
−1 −12 −2
]
4. Znaleźć macierz
X
spełniającą dane równanie macierzowe:
A
⋅X =B
A
=
[
2
1
−1 3
]
, B
=
[
4
5
]
A
⋅X =B⇒ X = A
−1
⋅B
D
11
=−1
1
1
⋅3=3
D
12
=−1
1
2
⋅1=−1
D
21
=−1
2
1
⋅1=−1
D
22
=−1
2
2
⋅2=2
A
D
=
[
3
1
−1 2
]
T
=
[
3
−1
1
2
]
A
−1
=
[
3
7
−
1
7
1
7
2
7
]
X
=
[
3
7
−
1
7
1
7
2
7
]
⋅
[
4
5
]
=
[
1
2
]