need help with both parts c AND d this is what i have for pa

need help with both parts (c) AND (d), this is what i have for parts (a) and (b):

3. Suppose that B and C are n x n matrices, and that N is an invertible n x n matrix so that CXN-1BN. (a) Find a formula expressing B in terms of N and C (i.e., \"Solve for B\"). (b) Show that C N-1B2 N. (HINT: just multiply.) c Suppose that D is a diagonal matrix with real entries. If we want to find a diagonal matrix C with real entries, such that C2 D, what has to be true about the eigenvalues of D? (d) Let A 16 10 50 29 Find a real matrix B with B A (i.e., a \"square root\" of A)

Solution

(a) If C = N^-1BN, then multiplying both the sides to the left by N and to the right by N^-1, we get NCN^-1 = N(N^-1BN)N^-1 = (NN^-1) B (NN^-1) = IBI = B. Thus, B = NCN^-1.

(b) We have C² = C.C = (N^-1BN)(N^-1BN)= N^-1B(NN^-1)BN = N^-1(BI)(B N)= N^-1(B)(B N)= N^-1(BB)N= N^-1B² N.

(c) If D is a diagonal matrix, the eigenvalues of D are its diagonal entries.

(d) The trace of A is T = -16+29 = 13.The determinant of A is D = (-16)(29)-(-10)(50)=-464+500= 36.Let a = D = ±36 = ±6 and b = ±(T+2a) = ±( 13±12) = ±25 = ±5 or, ±1= ±1. If B² = A, we have B = (1/b) C where C =

-16+a

-10

50

29+a

i.e. C =

-22

50

23

    Or, C =

-10

-10

50

35

-16+a	-10
50	29+a