Content deleted Content added
Citation bot (talk | contribs) Added bibcode. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Matrices (mathematics) | #UCB_Category 196/234 |
m Open access bot: url-access=subscription updated in citation with #oabot. |
||
Line 129:
Compound and adjugate matrices appear when computing determinants of [[linear combination]]s of matrices. It is elementary to check that if {{math|''A''}} and {{math|''B''}} are {{math|''n'' × ''n''}} matrices then
:<math>\det(sA + tB) = C_n\!\left(\begin{bmatrix} sA & I_n \end{bmatrix}\right)C_n\!\left(\begin{bmatrix} I_n \\ tB \end{bmatrix}\right).</math>
It is also true that:<ref>{{Cite journal|last1=Prells|first1=Uwe|last2=Friswell|first2=Michael I.|last3=Garvey|first3=Seamus D.|date=2003-02-08|title=Use of geometric algebra: compound matrices and the determinant of the sum of two matrices|url=http://rspa.royalsocietypublishing.org/content/459/2030/273|journal=Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences|language=en|volume=459|issue=2030|pages=273–285|doi=10.1098/rspa.2002.1040|bibcode=2003RSPSA.459..273P |s2cid=73593788 |issn=1364-5021|url-access=subscription}}</ref><ref>Horn and Johnson, p. 29</ref>
:<math>\det(sA + tB) = \sum_{r=0}^n s^r t^{n-r} \operatorname{tr}(\operatorname{adj}_r(A)C_r(B)).</math>
This has the immediate consequence
| |||