I M21I =
= a 12a33
— a13a 32
The minors of other elements of A can be found similarly.
The cofactor of the element aij of an N x N matrix A, denoted by cij, is defined as
In other words, a cofactor is a signed minor, the sign being positive if i + j is even and being negative if i + j is odd. Thus, the cofactor of the element a11 of the 3 x 3 matrix A given previously is a22a33 — a23a32, whereas the cofactor of the element a2i is —(ai2a33 — ai3a32) since the sum of the subscripts 2 and 1 is 3, which is an odd number.
Cofactor Matrix. Replacing the elements aij of a matrix A by their co-factors, we obtain a matrix known as the cofactor matrix of A, denoted by (cof A).
Adjoint Matrix. The adjoint matrix, written as (adj A), is the transpose of the cofactor matrix; that is, (adj A) = (cof A)'.
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