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A real n times n matrix A= aij is defined as aij={ i if i=j, 0 otherwise} then sum of all n eigen values of matrix a is?
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A real n times n matrix A= aij is defined as aij={ i if i=j, 0 otherwi...
Eigenvalues of Matrix A


A real n times n matrix A= aij is defined as aij={ i if i=j, 0 otherwise}. The task is to find the sum of all n eigenvalues of matrix a.


Eigenvalues and Eigenvectors


Eigenvalues and eigenvectors are an important part of linear algebra. Eigenvalues are scalar values that are associated with a matrix. An eigenvector is a non-zero vector that, when multiplied by a matrix, remains proportional to the original vector.


Finding Eigenvalues of Matrix A


To find the eigenvalues of matrix A, we need to find the values of λ for which the equation Ax=λx has non-zero solutions.


Let's solve the equation Ax=λx for matrix A:


Ax=λx


(aij)xj=λxi


aixi=λxi


aij=λ if i=j, 0 otherwise


Therefore, the eigenvalues of matrix A are λ1=1, λ2=2, ..., λn=n.


Calculating Sum of Eigenvalues


The sum of eigenvalues of matrix A is:


λ1+λ2+...+λn=1+2+...+n=n(n+1)/2


Therefore, the sum of all n eigenvalues of matrix A is n(n+1)/2.


Conclusion


In conclusion, the sum of all n eigenvalues of matrix A= aij is defined as aij={ i if i=j, 0 otherwise} is n(n+1)/2.
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A real n times n matrix A= aij is defined as aij={ i if i=j, 0 otherwise} then sum of all n eigen values of matrix a is?
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