Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.

Let $$\lambda_1,\lambda_2,\lambda_3,\lambda_4$$, and $$\lambda_5$$ be the five eigenvalues of A. Note that these eigenvalues need not be distinct.

The value of $$\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5=$$ ______________

Consider the following two statements with respect to the matrices A_{m $$\times$$ n ,} B_{n $$\times$$ m} , C_{n$$\times$$ n} and D_{n $$\times$$ n} .

Statement 1 : tr(AB) = tr(BA)

Statement 2 : tr(CD) = tr(DC)

where tr( ) represents the trace of a matrix. Which one of the following holds?

Suppose that P is a 4 × 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]T​. The rank of P is _________

Let X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?