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 multilinearity and the property to be alternating, directly for the determinant of a
3
×
3
{\displaystyle {}3\times 3}
-matrix.
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
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}}\,.}
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 satisfies 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.
Let
V
{\displaystyle {}V}
be a
vector space
over a
field
K
{\displaystyle {}K}
. For
a
∈
K
{\displaystyle {}a\in K}
,
the
linear mapping
φ
:
V
⟶
V
,
v
⟼
a
v
,
{\displaystyle \varphi \colon V\longrightarrow V,v\longmapsto av,}
is called homothety
(or dilation )
with
scaling 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}
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
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}}.}