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Unformatted text preview: i = j , form a matrix A by replacing row j of A with another
copy of row i .
Since A has proportional rows,
n 0 = det A = n aik Cik =
k =1 ajk Cik .
k =1 Outline The Classical Adjoint A Formula for the Inverse of a Matrix A Formula for the Inverse of a Matrix Theorem
If A is invertible, then
A−1 = 1
adj(A).
det A Cramer’s Rule Eigenvalues Outline The Classical Adjoint A Formula for the Inverse of a Matrix Cramer’s Rule The Proof
Proof.
Set B = adj(A) = [Cij ]T and AB = [cij ].
For each 1 ≤ i , j ≤ n,
n cij = aik Cjk =
k =1 det A if i = j
0
if i = j Hence,
1
cij =
det A
so 1
det A AB 1 if i = j
0 if i = j = In . We conclude that A−1 = 1
det A adj(A). Eigenvalues...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.
 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Determinant

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