Definition of the determinant
This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License.
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from lingeo import random_int_list
For $n\times n$ matrices $A$, the determinant $\det(A)$ is defined through the following rules.
(Note that this statement still holds even when $k = 0$.)
Note that the determinants for $2\times 2$ and $3\times 3$ matrices agree with this definition.
As a consequence, if a matrix $A$ is invertible and
can be written as the product a sequence of elementary matrices $F_1\cdots F_k$,
then $\det(A) = \det(F_1)\cdots\det(F_k)\det(I_n) = \det(F_1)\cdots\det(F_k)$.
In contrast, if $A$ is not invertible,
then $\det(A) = 0$.
In particular, this happens when
By definitions, $\det(A) = \Vol_C(A) = \Vol_R(A)$ for any matrix $A$.
Thanks to row operations, the definition of $\det(A)$ assigns at least a value to $\det(A)$.
However, maybe the rules assigns more than one values to it.
That is, the function might not be well-defined.
A matrix $A$ can be written as the product of different sequences of elementary matrices.
For example, one may write
for elementary matrices $F_1,\ldots, F_k$ and $E_1,\ldots, E_h$.
However, it is not yet clear by the definition
that $\det(F_1) \cdots \det(F_k) = \det(E_1) \cdots \det(E_h)$.
We will deal with this issue at the end of this chapter.
### code
set_random_seed(0)
print_ans = False
n = 4
while True:
A = matrix(n, random_int_list(n^2, 3))
if A.det() != 0:
break
print("A =")
pretty_print(A)
if print_ans:
print("determinant of A =", A.det())
將 $A$ 消成最簡階梯形式、
並記錄下每一步的列運算。
Find the reduced echelon form of $A$ and record each step of the row operation.
對以下矩陣 $A$,
求出 $\det(A)$。
提示:把所有列加到第一列。
For the following matrices $A$, find $\det(A)$.
Hint: Add each row to the first row.
令 $A$ 為一 $n\times n$ 矩陣
$$ \begin{bmatrix} n & -1 & \cdots & -1 \\ -1 & n & \ddots & \vdots \\ \vdots & \ddots & \ddots & -1 \\ -1 & \cdots & -1 & n \end{bmatrix}. $$Let $A$ be the $n\times n$ matrix
$$ \begin{bmatrix} n & -1 & \cdots & -1 \\ -1 & n & \ddots & \vdots \\ \vdots & \ddots & \ddots & -1 \\ -1 & \cdots & -1 & n \end{bmatrix}. $$利用 $\det$ 定義中的四條準則,說明以下性質。
Use the four rules in the definition of the determinant to explain the following properties.
若 $A$ 中的列向量集合線性相依,說明 $\det(A) = 0$。
If some rows of $A$ form a linearly dependent set, then $\det(A) = 0$.