A real n × n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0 The summation of all n eigen values of A is A. \[\frac{{{\text{n}}\left( {{\text{n}} + 1} \right)}}{2}\] B. \[\frac{{{\text{n}}\left( {{\text{n}} – 1} \right)}}{2}\] C. \[\frac{{{\text{n}}\left( {{\text{n}} + 1} \right)\left( {2{\text{n}} + 1} \right)}}{6}\] D. \[{{\text{n}}^2}\]

” option2=”\[\frac{{{\text{n}}\left( {{\text{n}} – 1} \right)}}{2}\]” option3=”\[\frac{{{\text{n}}\left( {{\text{n}} + 1} \right)\left( {2{\text{n}} + 1} \right)}}{6}\]” option4=”\[{{\text{n}}^2}\]” correct=”option4″]

The correct answer is $\boxed{\frac{{{\text{n}}\left( {{\text{n}} + 1} \right)}}{2}}$.

A real n × n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0. This means that the matrix is a diagonal matrix with all the diagonal elements equal to 1.

The sum of all the eigenvalues of a diagonal matrix is equal to the sum of all the diagonal elements. In this case, the sum of all the diagonal elements is n, so the sum of all the eigenvalues is $\frac{{{\text{n}}\left( {{\text{n}} + 1} \right)}}{2}$.

Option A is incorrect because it is the sum of the first n natural numbers. Option B is incorrect because it is the sum of the first n even natural numbers. Option C is incorrect because it is the sum of the first n odd natural numbers. Option D is incorrect because it is the square of n.