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There are many useful applications of the determinant. Cofactor expansion is one technique in computing determinants.

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There are many useful applications of the determinant. Cofactor expansion is one technique in computing determinants. Now, we discuss how to find these cofactors through minors of a matrix and use both of these elements to find the adjoint of A. Then by the adjoint and determinant, we can develop a formula for finding the inverse of a matrix. Minors: To find the minors of any matrix, expand block out every row and column one at a time until all the minors are found. Steps to Finding Each Minor Of A Matrix: 1. Delete the i
th
row and j
th
column of the matrix. 2. Compute the determinant of the remaining matrix after deleting the row and column of step 1. Example: Find the minors of the matrix
1 1 1 2 1 1 1 1 2 . *Note: This step procedure just outlines finding the minor M
11
of the matrix. 1. Delete the i
th
row and j
th
column of the matrix.
1 1 1 2 1 1 1 1 2 2. Compute the determinant of the remaining matrix after deleting the row and column of step 1. ( 1)( 1) (1)(1) 0
1 1 1 1 det 1 1 1 2 1 1 1 1 2
11
M Using the same steps above, the other minors of the matrix are given below. 3 1 1 2 1 det
12
M
3 1 1 2 1 det
13
M
1 1 1 1 2 det
21
M
3 1 1 1 2 det
22
M
2 1 1 1 1 det
23
M
1 1 1 1 2 det
31
M
3 2 1 1 2 det
32
M
1 2 1 1 1 det
33
M
Thus, the minor matrix is given by
1 3 1 1 3 2 0 3 3 M Cofactors: To find the cofactors of a matrix, just use the minors and apply the following formula: C
ij
= (-1)
i + j
M
ij
where M
ij
is the minor in the i
th
row, j
th
position of the matrix. Example: Find the cofactors of the matrix

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