Note that =0. One of a number of largest submatrices whose determinant is not
equal to zero is
Hence, the rank of the matrix Ais 3.
MinorMij. If the ith row and jth column are deleted from an n*nmatrixA,
the resulting matrix is an(n-1)*(n-1) matrix. The determinant of this
(n-1) (n-1)matrix is called the minor Mijof the matrix A.
CofactorAij. The cofactor Aijof the element aijof the n*nmatrixAis defined
by the equation
Aij=(1)ijMij
That is, the cofactor Aijof the element aijis(1)ijtimes the determinant of the matrix
formed by deleting the ith row and the jth column from A. Note that the cofactor Aijof
the element aijis the coefficient of the term aijin the expansion of the determinant ,
since it can be shown that
If are replaced by then
because the determinant of Ain this case possesses two identical rows. Hence, we obtain
Similarly,
Adjoint Matrix. The matrix Bwhose element in the ith row and jth column equals
Ajiis called the adjoint of Aand is denoted by adj A,or
B=(bij)=(Aji)=adjA
That is, the adjoint of Ais the transpose of the matrix whose elements are the cofactors
ofA,or
adjA= D
A 11 A 21 p An 1
A 12 A 22 p An 2
oo o
A 1 n A 2 n p Anm
T
a
nk= 1akiAkj=dij@A@
a
nk= 1ajkAik=dji@A@
aj 1 Ai 1 +aj 2 Ai 2 +p+ajnAin= 0 iZj
ai 1 ,ai 2 ,p,ain aj 1 ,aj 2 ,p,ajn,
ai 1 Ai 1 +ai 2 Ai 2 +p+ainAin=@A@
@A@
C
1 2 3
01 - 1
10 1
S
@A@
876 Appendix C / Vector-Matrix AlgebraOpenmirrors.com