made easy test series que

Question

a real n*n matrix aij =i if i = j , otherwise 0

the determinant of all n eigen values of A is

a) n(n+1)/2

b) n(n-1)/2

c) n(n+1)(2n+1)/6

d) factorial(n)

my ans is d but the ans given is a. Is it correct?

Comments

It is not like that.The product of eigen values of matrix A = |A|

There is nothing like determinant of eigen value.It is meaningless.

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i think quetion is for determinant of matrix A = which is product of eigen value of  matrix A = n!

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Yes in that case it will be n! since it is a diagonal matrix , hence product of eigen values = determinant of matrix A = product of diagonal values(since they are eigen values themselves) which is hence given by,

1.2.3......n = n!

Answer 1

For any nxn matrix, given with the conditions as..
For i=j, aij=i..
And 0 otherwise....!

So, this will lead out to the diagonal matrix, with the entries as:
1,2,3,4,5...........n...!!

And for the diagonal matrix, the eigen values are the diagonal entries themselves..!!
Hence, we know that the determinant of the diagonal matrix will be the product of all the diagonal entries...!!

Which will come out to..
1 x 2 x 3 x 4 x.........x n = n!