Adjoints and Determinants Problem

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SUMMARY

The discussion focuses on the relationship between adjoints and determinants in matrix algebra. The formula A-1 = [1/det(A)]adj(A) is highlighted as essential for understanding matrix inverses. Additionally, it is established that det(adj(A)) = det(A)n-1, where n represents the size of matrix A. The adjoint of A is defined as the matrix of cofactors, which is crucial for calculating determinants through cofactor expansion.

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  • Understanding of matrix algebra concepts
  • Familiarity with determinants and their properties
  • Knowledge of cofactors and adjoint matrices
  • Basic skills in linear algebra
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  • Study the properties of determinants in linear algebra
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  • Investigate the implications of the formula det(adj(A)) = det(A)n-1
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I need help with this problem, i am totally lost. See attachment. Please explain.
 

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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
 
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)
 

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