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.