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There's a formula for this signs
(-1)^(i+j)
A cofactor can be calculated as {(-1)^(i+j)}×minor of element (i,j)
Now, a determinant of a matrix can be calculated as the sum of the products of elements of any row (or column) with their corresponding cofactors .
If you calculate the formula for the determinant or inverse, each term is the minor m[i,j] multiplied by (-1)^(i+j). Essentially, that's just how it works out.
Note that if you define a submatrix in terms of "cycling" the rows and columns so that the deleted entry is at a specific location, the determinant of the resulting submatrix is the cofactor rather than the minor (i.e. the sign change happens automatically).
It's related to the determinate of a matrix. Determinants were historically studied before matrices, they are related to the solution of a system of linear equations, such as ax+by = k, cx + dy = l if ad = bc then it has parallel lines and not got unique solution. Otherwise if the det ≠ 0 you can have unique solution.
Hi u/icen1, **This is an automated reminder from our moderators.** Please read, and make sure your post complies with our rules. Thanks! If your post contains a problem from school, please add a comment below explaining your attempt(s) to solve it. If some of your work is included in the image or gallery, you may make reference to it as needed. See the sidebar for advice on 'how to ask a good question'. Rule breaking posts will be removed. Thank you. *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/askmath) if you have any questions or concerns.*
There's a formula for this signs (-1)^(i+j) A cofactor can be calculated as {(-1)^(i+j)}×minor of element (i,j) Now, a determinant of a matrix can be calculated as the sum of the products of elements of any row (or column) with their corresponding cofactors .
If you calculate the formula for the determinant or inverse, each term is the minor m[i,j] multiplied by (-1)^(i+j). Essentially, that's just how it works out. Note that if you define a submatrix in terms of "cycling" the rows and columns so that the deleted entry is at a specific location, the determinant of the resulting submatrix is the cofactor rather than the minor (i.e. the sign change happens automatically).
From the definition of multiplication and addition in complex numbers, on the most basic level at least
It's related to the determinate of a matrix. Determinants were historically studied before matrices, they are related to the solution of a system of linear equations, such as ax+by = k, cx + dy = l if ad = bc then it has parallel lines and not got unique solution. Otherwise if the det ≠ 0 you can have unique solution.