ALL-ZEBRAS

Here are some Zebras to Hunt for !!

Determine the value of the determinant of \[\begin{vmatrix}\gcd (1,1)&\gcd (1,2)&\cdots &\gcd (1,n)\\\gcd (2,1)&\gcd (2,2)&\cdots &\gcd (2,n)\\\vdots&\vdots&\ddots&\vdots\\\gcd (n,1)&\gcd (n,2)&\cdots &\gcd (n,n)\end{vmatrix}\]


Distinct Eigen-Values correspond to linearly independent Eigen-Vectors.(Prove it !!)

If all eigen vectors are linearly independent then the matrix is diagonalizable and the diagonal matrix has \(\lambda_1 , \lambda_2, \cdots \lambda_n\) as diagonal entries.

Let \(a_n = (\ln 3)^n \sum_{k=1}^{n} \frac{k^2}{k!(n-k)!}\) . Then the sum of the series \(a_1 + a_2 + a_3 + ... \to \infty\) , is equal to?

A is a \(n \times m\) matrix and B is a \(m \times n\) matrix . If \(|I_{n} - AB|\) is non-singular so is \(|I_{m} - BA|\)

For each positive integer \(n \ge 3\), define \(A_n\) and \(B_n\) as \(A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\) , \(B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\) Determine all positive integers \(n\ge 3\) for which \(\lfloor A_n \rfloor = \lfloor B_n \rfloor\).

For any real number \(x\), \(\lfloor x\rfloor\) denotes the largest integer \(N\le x\).