Adjoints and Determinants Problem

In summary, the conversation discusses the topic of computing the determinant and adjoint of a matrix. It is suggested to use the formula A-1=[1/det(A)]adj(A) and to consider the relationship between cofactors and the adjoint. It is also mentioned that the adjoint of A is simply the matrix of cofactors.
  • #1
CanadianEh
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I need help with this problem, i am totally lost. See attachment. Please explain.
 

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  • #2
What have you tried so far? Perhaps use the famous formula that A-1=[1/det(A)]adj(A).
Also note that det(adj(A)) = det(A)n-1 where n is the size of A
 
  • #3
You need to think about what the adjoint of A is...

The adjoint of A is simply the matrix of cofactors.

So if you took matrix A and did an expansion across the 3rd row to compute its determinant what are you really doing?

Well you're taking the entries 3, -1, -1, multiplying each of them by their respected cofactors and summing them to get the determinant.

How can you relate the desired cofactors to the adjoint? (I stated the answer above)
 

Related to Adjoints and Determinants Problem

1. What are adjoints and determinants?

Adjoints and determinants are mathematical concepts used in linear algebra. The adjoint of a matrix is a matrix that is obtained by taking the transpose of the cofactor matrix of the original matrix. The determinant of a matrix is a scalar value that represents the scaling factor of the linear transformation represented by the matrix.

2. How are adjoints and determinants related?

The adjoint and determinant of a matrix are closely related, as the determinant can be calculated using the adjoint matrix. The determinant of a matrix is also equal to the product of its eigenvalues, which can be found using the adjoint matrix.

3. What is the purpose of using adjoints and determinants?

Adjoints and determinants have various applications in mathematics and science. They are used to solve systems of linear equations, calculate the inverse of a matrix, and find the area of a parallelogram or volume of a parallelepiped. They are also used in physics to calculate moments of inertia and in engineering to solve optimization problems.

4. How are adjoints and determinants calculated?

Calculating the adjoint and determinant of a matrix involves several steps. To find the adjoint, the cofactor matrix of the original matrix must be calculated, and then the transpose of this matrix is taken. To find the determinant, there are several methods such as using cofactor expansion, Gaussian elimination, or the Leibniz formula.

5. Can adjoints and determinants be used for non-square matrices?

No, adjoints and determinants can only be calculated for square matrices. The number of columns must be equal to the number of rows in order for the adjoint and determinant to be defined. However, they can be extended to non-square matrices by using the generalized adjoint and determinant, which are used in the study of abstract algebra.

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