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 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 multi-linearity and the property to be alternating, directly for the determinant of a
3
×
3
{\displaystyle {}3\times 3}
-matrix.
===Exercise Exercise 26.9
change ===
Let
M
{\displaystyle {}M}
be the following square matrix
M
=
(
A
B
0
D
)
,
{\displaystyle {}M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,}
where
A
{\displaystyle {}A}
and
D
{\displaystyle {}D}
are square matrices. Prove that
det
M
=
det
A
⋅
det
D
{\displaystyle {}\det M=\det A\cdot \det D}
.
Determine for which
x
∈
C
{\displaystyle {}x\in \mathbb {C} }
the matrix
(
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})}
and
(
x
2
,
y
2
)
{\displaystyle {}(x_{2},y_{2})}
,
the determinant of the
2
×
2
{\displaystyle {}2\times 2}
-matrix defined by these vectors is equal
(up to sign)
to the area of the plane parallelogram spanned by the vectors.
Let
K
{\displaystyle {}K}
be a field and
n
∈
N
+
{\displaystyle {}n\in \mathbb {N} _{+}}
.
Show that the
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\}}
and arbitrary
n
−
1
{\displaystyle {}n-1}
vectors
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}}
,
for
u
∈
K
n
{\displaystyle {}u\in K^{n}}
and for
s
∈
K
{\displaystyle {}s\in K}
)
the equality
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}
be a field and
m
,
n
,
p
∈
N
{\displaystyle {}m,n,p\in \mathbb {N} }
.
Prove that the transpose of a matrix satisfy the following properties (where
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)}
and
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}
-matrices, such that in each column and in each row there are exactly one
1
{\displaystyle {}1}
and two
0
{\displaystyle {}0}
s.
Let
z
∈
C
{\displaystyle {}z\in \mathbb {C} }
and let
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}}
.
The next exercises use the following definition.
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
===Exercise (m+ marks) Exercise 26.22
change ===
Let
K
{\displaystyle {}K}
be a field, and let
V
{\displaystyle {}V}
and
W
{\displaystyle {}W}
be vector spaces over
K
{\displaystyle {}K}
of dimensions
n
{\displaystyle {}n}
and
m
{\displaystyle {}m}
. Let
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)}
with respect to two bases. Prove that
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}}.}