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.
Show that the elementary operations on the rows do not change the column rank.
Determine the
determinant
of a
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 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
of the
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 multi-linearity and the property to be alternating, directly for the determinant of a
3
×
3
{\displaystyle {}3\times 3}
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
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}}\,.}
Prove that you can expand the determinant according to each row and each column.
Let
K
{\displaystyle {}K}
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}}
Let
V
{\displaystyle {}V}
vector space
over a
field
K
{\displaystyle {}K}
a
∈
K
{\displaystyle {}a\in K}
linear mapping
φ
:
V
⟶
V
,
v
⟼
a
v
,
{\displaystyle \varphi \colon V\longrightarrow V,v\longmapsto av,}
is called homothety
(or dilation )
with
scale factor
a
{\displaystyle {}a}
.
What is the determinant of a 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 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
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}}.}
