Understanding Cramer's Rule for Inverse Matrices
Explore Cramer's Rule, a method for finding the inverse of a matrix, through detailed formulas, proofs, and applications. Learn how to calculate the adjugate matrix and use it to determine the inverse of a matrix systematically.
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Presentation Transcript
Formula for A-1 ?11 ??1 ?1? ??? 1 ? 1= ??? ? : scalar ?: cofactors of A (? has the same size as A, so does ??) ??is adjugate of A (adj A, ) ??? ??? ? = ?11 ?21 ? ? ? ? ?12 ?22 ? ? ? ? ? = ? =? ? ? ? 1 ? = 1 ? ? ? ??? ? = ? ?? ?? ??= = ?? ??
1 Formula for A-1 ? 1= ??? ??? ? ? ? ? ? ? ? ? ,? 1=? ? = ??? ? = ??? + ??? + ?? ??? ??? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? ? = + ? ? ? ? ? ? ? ? ? ? ? ? + +
1 Formula for A-1 ? 1= ??? ??? Proof: ???= ??? ? ?? ?11 ??1 ?1? ??? ?11 ?1? ??1 ??? ??? ? 0 = 0 ??? ? transpose Diagonal: By definition of determinants Not Diagonal: (Exercise 82, P221)
???= ??? ? ?? ?11 ??1 ?1? ??? ?11 ?1? ??1 ??? ??? ? 0 = 0 ??? ? ?11 ??1 ?1? ??? = ?11?11+ ?12?12+ + ?1??1? ??? ??1 ??1 = 0 ??? ??? = ??1?11+ ??2?12+ + ????1? ???
?11 ?1? ??1 ??? Cramer s Rule 1 1 ? = ? 1? = ??? ???? ? 1= ??? ??? ?? = ? 1 ?1= ?11?1+ ?21?2+ + ??1?? ??? ?1 ?1=??? ?1 ??? ? ??? ? ?2=??? ?2 ??? ? ?1= with column 1 replaced by b n-1 Columns of A ? ??=??? ?? ??? ? ??= with column j replaced by b
