True or False For all subquestions below assume that A is an

True or False.

For all subquestions below, assume that A is an n times n real matrix. T F: lambda is an eigenvalue of A if and only if - lambda is an eigenvalue of -A. T F: If A is an n times n matrix and lambda is one of its eigenvalues, then rank(A lambda I_n) 4det (A) > 0 and tr(A)

Solution

1.     We know that is an eigenvalue of A if and only if Av = v where v 0 is an eigenvector of A.Thus, if Av = v , then (-A)v= –v so that – is an eigenvalue of –A. The statement is true.

        2. If v is an eigenvector of A then v 0 and Av = v for some scalar . Hence (A – I_n)v = 0.

This implies that the nullity of A I is positive. Hence rank(A – I_n) < n. The statement is true.

  

       3. The eigenvalues of a 2x2 matrix A are ₁ , ₂ = ½[ tr(A) ± { tr(A)² 4Det(A)}]. If tr(A)² > 4Det(A), then {tr(A)² 4Det(A)} is real, but smaller than tr(A). Thus, tr(A) < 0, then both eigenvalues will be real and negative . The statement is true.

4. If = 0 is not an eigenvalue of A, then A is invertible so that the RREF of A has no zero row or column. Then the column space of A is Rⁿ. The statement is true.

     5. If the column vectors of A ( a nxn matrix) are linearly independent, then A is invertible and hence = 0 is not an eigenvalue of A. The statement is true.