Exercises Determine explicitly the column rank and the row rank of the matrix
(
3
2
6
4
1
5
6
−
1
3
)
.
{\displaystyle {\begin{pmatrix}3&2&6\\4&1&5\\6&-1&3\end{pmatrix}}.}
Describe linear dependencies (if they exist) between the rows and between the columns of the matrix.
Exercise Exercise 26.2
change ===
Show that the elementary operations on the rows do not change the column rank.
Determine the
determinant MDLD/determinant
of a
plane rotation.MDLD/plane rotation
Compute the determinant of the matrix
(
1
+
3
i
5
−
i
3
−
2
i
4
+
i
)
.
{\displaystyle {\begin{pmatrix}1+3{\mathrm {i} }&5-{\mathrm {i} }\\3-2{\mathrm {i} }&4+{\mathrm {i} }\end{pmatrix}}.}
Compute the
determinant MDLD/determinant
of the matrix
(
1
3
5
2
1
3
8
7
4
)
.
{\displaystyle {\begin{pmatrix}1&3&5\\2&1&3\\8&7&4\end{pmatrix}}.}
Compute the
determinant MDLD/determinant
of the
matrix MDLD/matrix
(
1
3
9
0
−
1
0
5
2
0
1
3
6
−
3
0
0
7
)
.
{\displaystyle {\begin{pmatrix}1&3&9&0\\-1&0&5&2\\0&1&3&6\\-3&0&0&7\end{pmatrix}}.}
Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.
Check the multilinearity and the property to be alternating, directly for the determinant of a
3
×
3
{\displaystyle {}3\times 3}
Exercise Exercise 26.9
change ===
Let
M
{\displaystyle {}M}
M
=
(
A
B
0
D
)
,
{\displaystyle {}M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,}
where
A
{\displaystyle {}A}
D
{\displaystyle {}D}
det
M
=
det
A
⋅
det
D
{\displaystyle {}\det M=\det A\cdot \det D}
Determine for which
x
∈
C
{\displaystyle {}x\in \mathbb {C} }
(
x
2
+
x
−
x
−
x
3
+
2
x
2
+
5
x
−
1
x
2
−
x
)
{\displaystyle {\begin{pmatrix}x^{2}+x&-x\\-x^{3}+2x^{2}+5x-1&x^{2}-x\end{pmatrix}}}
is invertible.
Use the image to convince yourself that, given two vectors
(
x
1
,
y
1
)
{\displaystyle {}(x_{1},y_{1})}
(
x
2
,
y
2
)
{\displaystyle {}(x_{2},y_{2})}
2
×
2
{\displaystyle {}2\times 2}
Let
K
{\displaystyle {}K}
n
∈
N
+
{\displaystyle {}n\in \mathbb {N} _{+}}
determinant MDLD/determinant
Mat
n
(
K
)
=
(
K
n
)
n
⟶
K
,
M
⟼
det
M
,
{\displaystyle \operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,}
fulfills
(for arbitrary
k
∈
{
1
,
…
,
n
}
{\displaystyle {}k\in \{1,\ldots ,n\}}
n
−
1
{\displaystyle {}n-1}
v
1
,
…
,
v
k
−
1
,
v
k
+
1
,
…
,
v
n
∈
K
n
{\displaystyle {}v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}}
u
∈
K
n
{\displaystyle {}u\in K^{n}}
s
∈
K
{\displaystyle {}s\in K}
det
(
v
1
⋮
v
k
−
1
s
u
v
k
+
1
⋮
v
n
)
=
s
det
(
v
1
⋮
v
k
−
1
u
v
k
+
1
⋮
v
n
)
.
{\displaystyle {}\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,.}
Exercise Exercise 26.13
change ===
Prove that you can expand the determinant according to each row and each column.
Let
K
{\displaystyle {}K}
field ,MDLD/field
and
m
,
n
,
p
∈
N
{\displaystyle {}m,n,p\in \mathbb {N} }
A
,
B
∈
Mat
m
×
n
(
K
)
{\displaystyle {}A,B\in \operatorname {Mat} _{m\times n}(K)}
C
∈
Mat
n
×
p
(
K
)
{\displaystyle {}C\in \operatorname {Mat} _{n\times p}(K)}
s
∈
K
{\displaystyle {}s\in K}
(
A
tr
)
tr
=
A
.
{\displaystyle {}{({A^{\text{tr}}})^{\text{tr}}}=A\,.}
(
A
+
B
)
tr
=
A
tr
+
B
tr
.
{\displaystyle {}{(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.}
(
s
A
)
tr
=
s
⋅
A
tr
.
{\displaystyle {}{(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.}
(
A
∘
C
)
tr
=
C
tr
∘
A
tr
.
{\displaystyle {}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.}
Compute the determinant of the matrix
(
0
2
7
1
4
5
6
0
3
)
,
{\displaystyle {\begin{pmatrix}0&2&7\\1&4&5\\6&0&3\end{pmatrix}},}
by expanding the matrix along every column and along every row.
Compute the determinant of all the
3
×
3
{\displaystyle {}3\times 3}
1
{\displaystyle {}1}
0
{\displaystyle {}0}
Let
z
∈
C
{\displaystyle {}z\in \mathbb {C} }
C
⟶
C
,
w
⟼
z
w
,
{\displaystyle \mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,}
be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map
R
2
→
R
2
{\displaystyle {}\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}}
What is the determinant of a
homothety ?MDLD/homothety
Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.
Check the multiplication theorem for determinants of the following matrices
A
=
(
5
7
2
−
4
)
and
B
=
(
−
3
1
6
5
)
.
{\displaystyle A={\begin{pmatrix}5&7\\2&-4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-3&1\\6&5\end{pmatrix}}.}
Confirm
the Multiplication theorem for determinants
for the matrices
A
=
(
1
4
1
1
2
0
0
1
1
)
and
B
=
(
2
0
1
0
1
0
1
0
1
)
.
{\displaystyle A={\begin{pmatrix}1&4&1\\1&2&0\\0&1&1\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}2&0&1\\0&1&0\\1&0&1\end{pmatrix}}.}
Hand-in-exercises ===Exercise (m+ marks) Exercise 26.22
change ===
Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
W
{\displaystyle {}W}
K
{\displaystyle {}K}
n
{\displaystyle {}n}
m
{\displaystyle {}m}
φ
:
V
⟶
W
{\displaystyle \varphi \colon V\longrightarrow W}
be a linear map, described by the matrix
M
∈
Mat
m
×
n
(
K
)
{\displaystyle {}M\in \operatorname {Mat} _{m\times n}(K)}
rk
φ
=
rk
M
.
{\displaystyle {}\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,.}
Compute the determinant of the matrix
(
1
+
i
3
−
2
i
5
i
1
3
−
i
2
i
−
4
−
i
2
+
i
)
.
{\displaystyle {\begin{pmatrix}1+{\mathrm {i} }&3-2{\mathrm {i} }&5\\{\mathrm {i} }&1&3-{\mathrm {i} }\\2{\mathrm {i} }&-4-{\mathrm {i} }&2+{\mathrm {i} }\end{pmatrix}}.}
Compute the determinant of the matrix
A
=
(
2
1
0
−
2
1
3
3
−
1
3
2
4
−
3
2
−
2
2
3
)
.
{\displaystyle {}A={\begin{pmatrix}2&1&0&-2\\1&3&3&-1\\3&2&4&-3\\2&-2&2&3\end{pmatrix}}\,.}
Check the multiplication theorem for the
determinants MDLD/determinants
of the following matrices
A
=
(
3
4
7
2
0
−
1
1
3
4
)
and
B
=
(
−
2
1
0
2
3
5
2
0
−
3
)
.
{\displaystyle A={\begin{pmatrix}3&4&7\\2&0&-1\\1&3&4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-2&1&0\\2&3&5\\2&0&-3\end{pmatrix}}.}
