标签:Intorduction AAA Linear nn detA frac cdots amp Determinants
Determinants
introduction to determinants
detA=∑j=1n(−1)i+ja1jAnj
detA=∑i=1n(−1)i+jai1Ain
if A is a triangular matrix the determinants is products of the entries on the main diagonal of A
Properties of determinants
Row Operations:
Let A be a square matrix.
a.If a multiple of one row of A is added to another row to produce a matrix B, then detB=detA.
b.If two rows of A are interchanged to produce B, then detB=−detA.
c.If one row of A is multipled by k to produce B, then detB=k∗detA.
Theorem:
If A is an n×n matrix, then detAT=detA
If A and B are n×n matrix, then det(AB)=detA×detB
Linearity Properties of the determinant Function
A(x)=[a1⋯ai−1xai+1⋯an]
T(x)=det[a1⋯ai−1xai+1⋯an]
T(cx)=cT(x)
T(a+b)=T(a)+T(b)
Cramer’s Rule:
Let A be an ivertible n×n matrix. For any b in Rn, the unique solution of Ax=b is:
x=detAdetAi(b)
Thus, from AA−1=E we could get A−1=⎣⎢⎡detAdetA1(e1)⋮detAdetAne1⋯⋮⋯detAdetA1en⋮detAdetAnen⎦⎥⎤
A−1=detA1adjA
标签:Intorduction,AAA,Linear,nn,detA,frac,cdots,amp,Determinants 来源: https://blog.csdn.net/zbzhzhy/article/details/87900484
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