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Determinants

Matrices are frequently used when investigating the solutions of systems of linear equations. Decisive for the solubility and the possible number of solutions of such a system is a number, called the determinant $\det(\mbox{\textit{\textbf{A}}})$ or $\vert\mbox{\textit{\textbf{A}}}\vert$ of A, which can be calculated for any $(n\times n)$ square matrix A. In this section determinants are introduced and some of their laws are stated. Determinants are used to invert matrices and to calculate the volume of a unit cell in Subsections 2.6.1 and 2.6.3.

The theory of determinants is well developed and can be treated in a very general way. We only need determinants of (2 $\times$ 2) and (3 $\times$ 3) matrices and will discuss only these.

Definition (D 2.5.1)
Let \( \mbox{\textit{\textbf{A}}}=\left( \begin{array}{cc} A_{11} & A_{12} \\
A_{21} & A_{22} \end{array} \right) \) and \( \mbox{\textit{\textbf{B}}}=\left( \begin{array}{ccc} B_{11} & B_{12} & B_{13} \\ B_{21} &
B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \end{array} \right) \) be a

(2 $\times$ 2) and a (3 $\times$ 3) matrix. Then their determinants are designated by

\( \det(\mbox{\textit{\textbf{A}}})=\left\vert \begin{array}{cc} A_{11} & A_{12}...
...B_{21} &
B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \end{array} \right\vert \) and are defined by the equations

\begin{displaymath}\begin{array}{rcl} \det(\mbox{\textit{\textbf{A}}}) & = & A_{11}\, A_{22}-
A_{12}\,A_{21}\ \ \ \mbox{and}
\end{array} \end{displaymath} (2.5.1)
\begin{displaymath}\begin{array}{rcl} \det(\mbox{\textit{\textbf{B}}}) & = & B_{...
...- B_{12}\,B_{21}\,B_{33} -
B_{13}\,B_{22}\,B_{31}. \end{array}\end{displaymath} (2.5.2)

Let D be a square matrix. If $\det(\mbox{\textit{\textbf{D}}})\ne 0$ then the matrix D is called regular, if $\det(\mbox{\textit{\textbf{D}}})=0$, then D is called singular. Here only regular matrices are considered. The matrix W of an isometry W is regular because always $\det(\mbox{\textit{\textbf{W}}})=\pm 1$. In particular, $\det(\mbox{\textit{\textbf{I}}})=+1$ holds.

Remark. The determinant $\det(\mbox{\textit{\textbf{A}}})$ is equal to the fraction $\tilde{V}/{V}$, where $V$ is the volume of an original object and $\tilde{V}$ the volume of this object mapped by the affine mapping A. Isometries do not change distances, therefore they do not change volumes and $\det(\mbox{\textit{\textbf{W}}})=\pm 1$ holds.

The following rules hold for determinants of $(n\times n)$ matrices A. The columns of A will be designated for these rules by $\mbox{\textit{\textbf{A}}}_k,\ k=1,\ldots,n$.

  1. $\det(\mbox{\textit{\textbf{A}}}^{\mbox{\footnotesize {T}}})=\det(\mbox{\textit{\textbf{A}}})$; the determinant of a matrix is the same as that of the transposed matrix. Because of this rule the following rules, although formulated only for columns, also hold if formulated for rows.
  2. If one column of $\det(\mbox{\textit{\textbf{A}}})$ is a multiple of another column, $\mbox{\textit{\textbf{A}}}_{k}=\lambda \mbox{\textit{\textbf{A}}}_{j}$, then $\det(\mbox{\textit{\textbf{A}}})=0$. This implies that $\det(\mbox{\textit{\textbf{A}}})=0$ if two columns of A are equal.
  3. If a column $\mbox{\textit{\textbf{A}}}_k$ is the sum of two columns $\mbox{\textit{\textbf{B}}}_k$ and $\mbox{\textit{\textbf{C}}}_k$, $\mbox{\textit{\textbf{A}}}_k=\mbox{\textit{\textbf{B}}}_k+\mbox{\textit{\textbf{C}}}_k$, then $\det(\mbox{\textit{\textbf{A}}})=
\det(\mbox{\textit{\textbf{B}}})+\det(\mbox{\textit{\textbf{C}}})$, where B is the matrix which has all columns of A except that $\mbox{\textit{\textbf{A}}}_k$ is replaced by $\mbox{\textit{\textbf{B}}}_k$, and C is the matrix with all columns of A except that $\mbox{\textit{\textbf{A}}}_k$ is replaced by $\mbox{\textit{\textbf{C}}}_k$.
  4. Exchange of two columns, $\mbox{\textit{\textbf{A}}}_j\longrightarrow\mbox{\textit{\textbf{A}}}_k$ and $\mbox{\textit{\textbf{A}}}_{k}\longrightarrow\mbox{\textit{\textbf{A}}}_{j}$ of a determinant changes its sign.
  5. Adding to a column a multiple of another column does not change the value of the determinant:
    $\vert\mbox{\textit{\textbf{A}}}_1\ \mbox{\textit{\textbf{A}}}_2 \ldots \mbox{\t...
...
\ldots \mbox{\textit{\textbf{A}}}_k \ldots \mbox{\textit{\textbf{A}}}_n\vert.$
  6. Multiplication of all elements of a column with a number $\lambda$ results in the $\lambda$-fold value of the determinant:

    \( \vert\mbox{\textit{\textbf{A}}}_1\ \mbox{\textit{\textbf{A}}}_2 \ldots \lambd...
...\ldots \mbox{\textit{\textbf{A}}}_k \ldots \mbox{\textit{\textbf{A}}}_n\vert. \)
  7. \( \det(\mbox{\textit{\textbf{A}}}\,\mbox{\textit{\textbf{B}}})=\det(\mbox{\textit{\textbf{A}}})\,\det(\mbox{\textit{\textbf{B}}}) \), i.e. the determinant of a matrix product is equal to the product of the determinants of the matrices.
  8. $\det(\mbox{\textit{\textbf{A}}}^{-1})=(\det(\mbox{\textit{\textbf{A}}}))^{-1}$, for $\mbox{\textit{\textbf{A}}}^{-1}$ see Subsection 2.6.1.

Among these rules there are three procedures which do not change the value of the determinant:

(i)
transposition;
(ii)
an even number of exchanges of columns (or rows correspondingly), because
an odd number of exchanges changes the sign of the determinant; and
(iii)
adding to a column a multiple of another column (or rows correspondingly).
Examples to the rules; calculation of the determinants according to equation (2.5.2).
  1. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 & 2 \\
3 & 2 & 3 \end{array}=-4; \) \( \mbox{\textit{\textbf{A}}}^{^{\mbox{\footnotesize {T}}}}=\begin{array}{\vert ccc\vert} 1 & 2 & 3 \\ 1 & 2 & 2 \\
3 & 2 & 3 \end{array}=-4. \)
  2. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 2 & A \\ 1 & 2 & B \\
3 & 6 & C \end{array}=2C+6B+6A-6B-2C-6A=0. \)
  3. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 & 2 \\
3 & 2 & 3 \end{array}=-4; \) \( \mbox{\textit{\textbf{B}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3
\\ 2 & 2 & 0
\\ 3 & 2 & 2 \end{array}=-6; \) \( \mbox{\textit{\textbf{C}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 0
\\ 2 & 2 & 2
\\ 3 & 2 & 1 \end{array}=+2; \) \( -4 = -6 + 2. \)
  4. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 & 2 \\
3 & 2 & 3 \end{array}=-4; \) \( \mbox{\textit{\textbf{B}}}=\begin{array}{\vert ccc\vert} 1 & 3 & 1 \\
2 & 2 & 2 \\ 3 & 3 & 2 \end{array}=+4. \)
  5. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 &...
...{array}{\vert ccc\vert} 2 & 1 & 3 \\
4 & 2 & 2 \\ 5 & 2 & 3 \end{array}=-4. \)
  6. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 &...
...2\,\lambda & 2 \\ 3 & 2\,\lambda & 3 \end{array}=(-4)\,\lambda=-4\,
\lambda. \)
  7. \( \mbox{\textit{\textbf{A}}}=\begin{array}{\vert ccc\vert} 1 & 1 & 3 \\ 2 & 2 &...
... & 4 \\ 1 & 2 & 1 \\ 0 & 1 & 3
\end{array}=-1; \hspace{0.5em} (-4)\,(-1)=+4 \)

    is the product of the determinants.

    The determinant of the product AB is \( \begin{array}{\vert rrr\vert}
0 & 5 & 14 \\ 0 & 6 & 16 \\ \bar{1} & 7 & 23 \end{array}=+4. \)

  8. \( \vert\mbox{\textit{\textbf{A}}}\vert=\begin{array}{\vert rrr\vert} 1 & 1 & 3 ...
...vert}
-1/2 & -3/4 & 1\\ 0 &3/2 & \bar{1} \\ 1/2 & -1/4 & 0 \end{array}=-1/4. \)


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