Please answer from (b) to (e) 1. Fill in the blank. No partial credit! (2 pts ea

Question

Please answer from (b) to (e)

1. Fill in the blank. No partial credit! (2 pts each) atristhat satisfies the equaticn Al::??:1.then the dete (a) Let A be 2 x 2 matrix that satisfies the equation A minant IA- (b) {?a, 0, b, 1] | a, b E R} is a subspace of R4 (c) Let A, D and P be n × n matrices such that A-PDP-1. Then D: Solve for D in terms of the other two matrices) (d) Consider the matrix equation Av-w where 4 is an eigenvalue of the invertible matrix A and v . It follows that an eigenvalue of A-1 is (e) A harmonic oscillator z + br + x = 0 is critically damped when

Explanation / Answer

(b). Let the given set be denoted by S and let X = (a,0,b,1) and Y = (c,0,d,1) be 2 arbitrary vectors in S. Then X+Y = (a,0,b,1)+(c,0,d,1)= (a+c,0,b+d,2). It implies that X+Y ? S. Hence S is not closed under vector addition, and, therefore, S is not a subspace of R4.

(c ). If A = PDP-1, then AP = (PDP-1)P= PD(P-1 P) = PD In =PD and then P-1 AP =P-1(PD) = (P-1P)D = In D = D. Hence D = P-1 AP.

(d).We know that a matrix A has an eigenvalue ? if and only if A-1 has eigenvalue 1/?. Thus, if 4 is an eigenvalue of A, then ¼ is an eigenvalue of A-1.

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