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Minors and Cofactors

In the determinant
Δ = 65132.png ...(1)
if we leave the row and the column passing through the element aij then the second-order determinant thus obtained is called the minor of aij and it is denoted by Mij. Thus, we can get nine minors corresponding to the nine elements.
For example, in determinant (1), the minor of the element
a21 = 65126.png
In terms of the notation of minors if we expand the determinant along the first row, then
Δ = (–1)1+1a11M11 + (–1)1+2 a12M12 + (–1)1+3 a13M13
= a11M11a12M12 + a13M13
Similarly, expanding Δ along the second column, we have
Δ = –a12M12 + a22M22a32M32
The minor Mij multiplied by (–1)i+j is called the cofactor of the element aij. If we denote the cofactor of the element aij, by Cij, then the cofactor of aij is Cij(–1)i+jMij.
Cofactor of the element a21 is
C21 = (–1)2+1M21 = 65120.png
In terms of the notation of the cofactors, we have
Δ = a11C11 + a12C12 + a13C13
= a21C21 + a22C22 + a23C23
= a31C31 + a32C32 + a33C33
Also, a11C21 + a12C22 + a13C23 = 0, a11C31 + a12C32 + a13C33 = 0, etc.

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