Mr惡魔 asked in 科學數學 · 1 decade ago

這題矩陣怎麼解??教教我...

 1  2  A=  2  0        B=   1  2  1         1  -1              0  -1  1 Find the value of  tr(AB)這題要怎麼解 有人能教我嗎>"<

Update:

有証明

若A是m*n,B是n*m,則AB是m*m,且BA是n*n,則tr(AB)=tr(BA)

嗎??

2 Answers

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  • 1 decade ago
    Favorite Answer

    Definition of the trace:

    suppose M is a square matrix(n*n) ,then the trace of M is the sum of the diagonal entries, and denoted by tr(M)=M(11)+M(22)+M(33)+...+M(nn)

    此題目可能有誤,因為A是2*2,B是2*3,則AB是2*3,並非方陣,所以並無法求tr(AB)。

    [補充:trace性質]

    若A是m*n,B是n*m,則AB是m*m,且BA是n*n,則tr(AB)=tr(BA)

    2006-10-22 09:40:38 補充:

    剛剛仔細看了題目原來A是3*2,B是2*3所以AB是3*3,則AB=1 0 3 2 4 2 1 3 0所以tr(AB)=1+4+0=5

    2006-10-23 01:39:12 補充:

    proof:let A=[a(ij)] is F(m*n) & B=[b(ij)] is F(n*m) and suppose AB=C=[c(ij)] is F(m*m) & BA=D=[d(ij)], thentr(AB)=tr(C)=sum[c(ij), i=1..m]= sum{sum[a(ik)b(ki),k=1..n],i=1..m}=sum{sum[b(ki)a(ik),i=1..m],k=1..n}=sum[d(kk),k=1..n] =tr(D)=tr(BA)

    Source(s): 自己
  • 1 decade ago

    非方陣不能求trace

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