Why is det(C)=det(A)^(n-1) for cofactors and determinants?

Thread starter Unusualskill
Start date Oct 15, 2014
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Determinants

In summary, cofactors and determinants are mathematical concepts used in linear algebra to solve systems of equations and calculate the inverse of a matrix. A cofactor is a numerical value calculated based on the position of a specific element in a matrix, while a determinant is a single numerical value representing the entire matrix. Cofactors are calculated by finding the determinant of a smaller matrix formed by removing the row and column containing the element, while determinants are calculated using various methods. Some applications of cofactors and determinants include solving systems of equations in fields such as physics, engineering, and economics, as well as manipulating data in computer graphics and data analysis. Both cofactors and determinants can be negative, depending on the position of the element in the

Oct 15, 2014

Unusualskill

Any1 can explain to me why det(C)=det(A)^(n-1)
where A is n-by-n matrix and C is the matrix of cofactors of A.
I have been thinking, any 1 can help?thx!

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Oct 15, 2014

Erland

Science Advisor

This follows from the equation AC=(det A)I, or equivalently A^-1= (1/det A)C (if det A≠0).

What are cofactors and determinants?

Cofactors and determinants are mathematical concepts used in linear algebra to solve systems of equations and calculate the inverse of a matrix.

What is the difference between a cofactor and a determinant?

A cofactor is a numerical value that is calculated based on the position of a specific element in a matrix, while a determinant is a single numerical value that represents the entire matrix as a whole.

How are cofactors and determinants calculated?

Cofactors are calculated by finding the determinant of a smaller matrix formed by removing the row and column that contain the specific element. Determinants are calculated using various methods, such as using the cofactor expansion method or using the rule of Sarrus for 3x3 matrices.

What are some applications of cofactors and determinants?

Cofactors and determinants are used in various fields such as physics, engineering, and economics to solve systems of equations and find the inverse of a matrix. They are also used in computer graphics and data analysis to transform and manipulate data.

Can cofactors and determinants be negative?

Yes, both cofactors and determinants can be negative. The sign of a cofactor depends on the position of the element in the matrix, while the sign of a determinant depends on the number of row or column swaps needed to transform the matrix into its upper triangular form.