[KOCW 선형대수] 15강. 행렬의 판별식

제 15강. 행렬의 판별식

m×n$$ square matrix det(A)$$

• Some important properties of determinant

1. A−1${}^{\mathrm{}}$ exist when det(A)≠0$\mathrm{}$

역행렬 식에 판별식의 역수가 곱해지기 때문,

2. det(A)$$ equals the volume of a box in ℝn${\mathbb{}}^{}$ space¶

Ex) Jacobian

3. det(A)=$$ + or - Product of the Pivots

4. Cramer's rule for Ax=b$$

xi=det(Ai)det(A)${}_{}\frac{{}_{}}{}$

Ex)

{x+y=32x−y=3$$\begin{array}{l}\mathrm{}\\ \mathrm{}\mathrm{}\end{array}$$

위의 식을 행렬로 바꾸면

[121−1][xy]=[33]$$\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\end{array}$$

위와 같이 표현할 수 있다.

Ax=[331−1]$$\mathbf{}\mathbf{}\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}$$

x$$를 구할 때, 좌측 열을 모두 3을 바꿔준다.

x=det(Ax)det(A)=−6−3=2$$\frac{{}_{}}{}\frac{\mathrm{}}{\mathrm{}}\mathrm{}$$

y도 같은 방법으로 구한다.

이번엔 우측 열을 모두 3으로 바꿔준다.

Ay=[1233]$$\mathbf{}\mathbf{}\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}$$

y=det(Ay)det(A)=−3−3=1$$\frac{{}_{}}{}\frac{\mathrm{}}{\mathrm{}}\mathrm{}$$

Properties Of Determinant

determinant by 2×2$\mathrm{}\mathrm{}$ matrix

det=[acbd]=ad−bc$$\mathbf{}\mathbf{}\mathbf{}\begin{array}{cc}& \\ & \end{array}$$

Basic Properties

1. det(I)=1$\mathrm{}$

2. The determinant changes sign(부호) when two rows are exchanged

3. The determinant depends linearly on the first row

[a+a′cb+b′d]=[acbd]+[a′cb′d]$$\begin{array}{cc}{}^{}& {}^{}\\ & \end{array}\begin{array}{cc}& \\ & \end{array}\begin{array}{cc}{}^{}& {}^{}\\ & \end{array}$$

det(A+B)=det(A)+det(B)$$$$

[tactbd]=t[acbd]$$\begin{array}{cc}& \\ & \end{array}\mathbf{}\begin{array}{cc}& \\ & \end{array}$$

det(tA)=tdet(A)=tndet(A)$${}^{}$$

4. If the rows are equal, then det(A)=0$\mathrm{}$

5. Subtracting a multiple of one row from another row leaves the same determinant

[a−lccb−lcd]=[acbd]+[−lcc−ldd]$$\begin{array}{cc}& \\ & \end{array}\begin{array}{cc}& \\ & \end{array}\begin{array}{cc}& \\ & \end{array}$$

맨 오른쪽 식의 판별식은 사라지게 된다

6. If A$$ has a row of zeros, then det(A)=0$\mathrm{}$

7. If A$$ is triangular, det(A)=$$ Product of diagnoal entries

8. If A$$ is singular, det(A)=0$\mathrm{}$ non-invertible / A$$ is non-singular ( invertible ) det(A)≠0$\mathrm{}$

9. det(AB)=det(A)det(B)$$

det(A−1)=1det(A)${}^{\mathrm{}}\frac{\mathrm{}}{}$

(1) B=I$$

(2) row exchange of A$$

10. det(AT)=det(A)${}^{}$

PA=LDU$$

det(PA)=det(P)det(A)=det(L)det(D)det(U)$$

det(PA)T=det(AT)det(PT)=det(UT)det(DT)det(LT)${}^{}{}^{}{}^{}{}^{}{}^{}{}^{}$

=det(L)det(D)det(U)$$