1. Full correct solutions.

2. Please, if you’re not sure you can solve these questions, don’t bid.

3. Preferable solutions from the lectures material/theorems, but it’s not a requirement. any correct solution will do.

Side note: the material is based on the book “Linear Algebra in Action”. (You can check there ‘kinda’ similar questions and explanations)

(1) In class we showed that if A € Cnxn and A > 0, then

(det A)1/n = min

trace AC

:Ce Chx, C> 0 and det C =

=1}

n

(1)

(a) Show that the formula (1) can be reexpressed as

-1}

п

traceCÁC

(det A)1/n = min

: Cecnxn, C> 0 and det C =

(2)

(b) Show by example that the formula (1) is false if A is singular.

[REMARK: The point in (b) is to show that there does not exist a

matrix C in the indicated class that achieves the minimum.]

(2) Showed that if A e Cnxn, A 0 and A is singular, then there exists

a sequence of matrices Ck, k = 1,2,…, such that Ck0, det k = 1

and trace ACk < 1/k.
(3) Show by direct estimates that if AER***, A > 0 and b € R” then the

function

3 (Ax,x) – (b, x) with x

is convex on R.

(4) Let p(r) belong to the class C²(a,b) of real valued functions that have

continuous first and second derivatives the interval (a,b). Show that

(a) If a < a < x. then
4(x) = f(a) + (x - a)'(a) + (x - 5)"(s)ds for a