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Chapter 16: Problem 3

Hat eine Matrix \(\boldsymbol{A} \in \mathbb{R}^{n \times n}\) mit \(n \in 2 \mathbb{N}+1\) und \(A=-A^{\mathrm{T}}\) die Determinante \(0 ?\)

Short Answer

Expert verified
Answer: Yes, the determinant of an odd-dimensional square matrix A, with the property that A = -A^T, is equal to 0.

Step by step solution

01

Write down the known information

We are given that \(A \in \mathbb{R}^{n \times n}\) and \(n \in 2\mathbb{N} + 1\). This means that \(A\) is an odd-dimensional square matrix. We are also given that \(A = -A^{\mathrm{T}}\).
02

Calculate determinant of \(A\) and \(A^{\mathrm{T}}\)

We know that the determinant of a matrix is equal to the determinant of its transpose, i.e., \(\det(A) = \det(A^{\mathrm{T}})\).
03

Use the given relation \(A = -A^{\mathrm{T}}\)

Now, we use the given relation about the matrix \(A\) being a skew-symmetric matrix, i.e., \(A = -A^{\mathrm{T}}\). Taking the determinant of both sides of this relation, we have \(\det(A) = \det(-A^{\mathrm{T}})\).
04

Use properties of determinant

We know that \(\det(kA) = k^n \det(A)\), where \(k\) is a scalar and \(A\) is an \(n \times n\) matrix. In our case, \(k = -1\). Therefore, \(\det(-A^{\mathrm{T}}) = (-1)^n \det(A^{\mathrm{T}})\). Since \(n\) is odd, \((-1)^n = -1\). So, \(\det(-A^{\mathrm{T}}) = -\det(A^{\mathrm{T}})\).
05

Combine equations from Step 2 and Step 4

Combining equations from Step 2 and Step 4, we get \(\det(A) = -\det(A^{\mathrm{T}})\).
06

Use the fact that \(\det(A) = \det(A^{\mathrm{T}})\)

Since we know that \(\det(A) = \det(A^{\mathrm{T}})\), we can substitute it into the equation from Step 5, which gives us \(\det(A) = -\det(A)\).
07

Solve for \(\det(A)\)

Now we have an equation like this: \(\det(A) = -\det(A)\). It implies that \(2 \det(A) = 0\), which means \(\det(A) = 0\). So, the determinant of matrix \(A\) is indeed 0.

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Most popular questions from this chapter

Matrix der Form $$ M=\left(\begin{array}{ll} A & C \\ 0 & B \end{array}\right) \in \mathbb{K}^{n \times n} $$ wobei \(\mathbf{0} \in \mathbb{K}^{(n-m) \times m}\) die Nullmatrix ist und \(\boldsymbol{A} \in \mathbb{K}^{m \times m}, \boldsymbol{C} \in\) \(\mathbb{K}^{m \times(n-m)}, \boldsymbol{B} \in \mathbb{K}^{(n-m) \times(n-m)}\) sind. Zeigen Sie: \(\operatorname{det} \boldsymbol{M}=\operatorname{det} \boldsymbol{A} \operatorname{det} \boldsymbol{B}\).

Folgt aus der Invertierbarkeit einer Matrix \(\boldsymbol{A}\) stets die Invertierbarkeit der Matix \(\boldsymbol{A}^{\mathrm{T}}\) ?

Gegeben sind drei Matrizen \(\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \in \mathbb{R}^{3 \times 3} \mathrm{mit}\) der Eigenschaft \(\boldsymbol{A} \boldsymbol{B}=\boldsymbol{C}\), ausf眉hrlich $$ \left(\begin{array}{ccc} a_{11} & 2 & 3 \\ a_{21} & 1 & 3 \\ a_{31} & -1 & -2 \end{array}\right)\left(\begin{array}{lll} 2 & b_{12} & 1 \\ 0 & b_{22} & 2 \\ 0 & b_{32} & 2 \end{array}\right)=\left(\begin{array}{ccc} 2 & -3 & c_{13} \\ 4 & -3 & c_{23} \\ 0 & 0 & c_{33} \end{array}\right) $$ Man erg盲nze die unbestimmten Komponenten.

F眉r ein aus drei produzierenden Abteilungen bestehendes Unternehmen hat man durch praktische Erfahrung die folgenden Matrizen \(P \in \mathbb{R}^{3 \times 3}\) f眉r die Produktherstellung und \(\boldsymbol{R} \in \mathbb{R}^{3 \times 3}\) f眉r die Rohstoffverteilung ermittelt: $$ \boldsymbol{P}=\left(\begin{array}{ccc} 0.5 & 0.0 & 0.1 \\ 0.0 & 0.8 & 0.2 \\ 0.1 & 0.0 & 0.8 \end{array}\right) \text { und } \boldsymbol{R}=\left(\begin{array}{ccc} 1 & 1 & 0.3 \\ 0.3 & 0.2 & 1 \\ 1.2 & 1 & 0.2 \end{array}\right) $$ (a) Welche Nachfrage \(v\) kann das Unternehmen befriedigen, wenn die Gesamtproduktion \(g\) durch \(g=\left(\begin{array}{c}150 \\ 230 \\\ 140\end{array}\right)\) bei Auslastung aller Maschinen vorgegeben ist? Welcher Rohstoffverbrauch \(r\) f盲llt dabei an? (b) Durch eine Marktforschung wurde der Verkaufsvektor \(v=\) \(\left(\begin{array}{l}90 \\ 54 \\ 36\end{array}\right)\) ermittelt. Welche Gesamtproduktion \(g\) ist n枚tig, um diese Nachfrage zu befriedigen? Mit welchem Rohstoffverbrauch \(\boldsymbol{r}\) ist dabei zu rechnen? (c) Nun ist die Rohstoffmenge \(r=\left(\begin{array}{l}200 \\ 100 \\\ 200\end{array}\right)\) vorgegeben. Welche Gesamtproduktion \(g\) kann erzielt werden? Welche Nachfrage \(v\) wird dabei befriedigt?

Gilt f眉r invertierbare Matrizen \(A, B \in \mathbb{K}^{n \times n}\) $$ \operatorname{ad}(\boldsymbol{A} \boldsymbol{B})=\operatorname{ad}(\boldsymbol{B}) \operatorname{ad}(\boldsymbol{A}) ? $$
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