行列式值的定義

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

Main idea

For $n\times n$ matrices $A$, the determinant $\det(A)$ is defined through the following rules.

• $\det(I_n) = 1$.
• If $E$ is the elementary matrix of $\rho_i\leftrightarrow\rho_j$, then $\det(EA) = -\det(A)$ and we define $\det(E) = -1$.
• If $E$ is the elementary matrix of $\rho_i:\times k$, then $\det(EA) = k\det(A)$ and we define $\det(E) = k$.

(Note that this statement still holds even when $k = 0$.)

• If $E$ is the elementary matrix of $\rho_i:+k\rho_j$, then $\det(EA) = \det(A)$ and we define $\det(E) = 1$.

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

• $A$ has a zero row, or
• $A$ has repeated rows.

By definitions, $\det(A) = \Vol_C(A) = \Vol_R(A)$ for any matrix $A$.

Remark

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

$$ A = F_1 \cdots F_k = E_1 \cdots E_h $$

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.

Side stories

• well-defined
• permutation matrices

Experiments

Exercise 1

執行以下程式碼。

Run the code below.

### 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())

Exercise 1(a)

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.

Exercise 1(b)

求出 $\det(A)$。

Find $\det(A)$.

Exercises

Exercise 2

對以下矩陣 $A$，
求出 $\det(A)$。

For the following matrices $A$, find $\det(A)$.

Exercise 2(a)

$$ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix}. $$

Exercise 2(b)

$$ A = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}. $$

Exercise 2(c)

$$ A = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix}. $$

Exercise 2(d)

$$ A = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 3 & 0 & 0 \\ 4 & 0 & 0 & 0 \\ \end{bmatrix}. $$

Exercise 2(e)

$$ A = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 3 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ \end{bmatrix}. $$

Exercise 3

對以下矩陣 $A$，
求出 $\det(A)$。

For the following matrices $A$, find $\det(A)$.

Exercise 3(a)

$$ A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 4 \end{bmatrix}. $$

Exercise 3(b)

$$ A = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 2 & \cdots & 2 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 2 & \cdots & n \end{bmatrix}. $$

Exercise 4

對以下矩陣 $A$，
求出 $\det(A)$。

For the following matrices $A$, find $\det(A)$.

Exercise 4(a)

$$ A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \end{bmatrix}. $$

Exercise 4(b)

$$ A = \begin{bmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 4 & 8 & 16 \\ 1 & 3 & 9 & 27 & 81 \\ 1 & 4 & 16 & 64 & 256 \\ 1 & 5 & 25 & 125 & 625 \end{bmatrix}. $$

Exercise 5

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.

Exercise 5(a)

$$ A = \begin{bmatrix} 4 & -1 & -1 & -1 \\ -1 & 4 & -1 & -1 \\ -1 & -1 & 4 & -1 \\ -1 & -1 & -1 & 4 \end{bmatrix}. $$

Exercise 5(b)

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

$$ \begin{bmatrix} n & -1 & \cdots & -1 \\ -1 & n & \ddots & \vdots \\ \vdots & \ddots & \ddots & -1 \\ -1 & \cdots & -1 & n \end{bmatrix}. $$

Exercise 6

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.

Exercise 6(a)

若 $A$ 中有一列為零向量，說明 $\det(A) = 0$。

If $A$ has a zero row, then $\det(A) = 0$.

Exercise 6(b)

若 $A$ 中有兩列向量相等，說明 $\det(A) = 0$。

If $A$ has two rows in common, then $\det(A) = 0$.

Exercise 6(c)

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 $\det(A) = 0$.