Consider the matrix A = I – X(X'X)-¹X'. Prove that A. A = A. What restrictions need to be imposed on X and (X'X) for this to hold?

Q: [a11(t) a12(t) 1) Let A(T) = be differentiable. %3D Show that: d (det (A(t)) = det a 12 (t)] la21(t)…

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Q: (BH) Consider the following matrix in R5×5. * 0 0 0 0 0 0 0 0 0 * A = * * * where the *s are…

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Q: = (3, 6) 2. It is known that u = (-2,3) and v = relative to the inner product in R? which is derived…

A: 2 Given that u=−2,3 and v=3,6 And A=2134. We have to find the u and v. We know that…

Q: Write a'= -3x+y+ln(t), y'= t'a in matrix form as 18 31 and f(t) is 181 [*] -= P(1) [*] + f(t) where…

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Q: 4. For the matrix 1 X' = 1 1 1 4 -2 3 -5 compute P = X(X'X)-1X' and M =I- P (a) Verify that MP = (0.…

A: Consider the provided matrix, X'=11114-23-5For to find P=XX'X-1X' ,X=14 1-2 13 1-5

Q: Give the matrix associated with the quadratic form 10(x,)? – 5x,x2 +6(x,)².

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Q: 27. Let A be square n x n-matrix. Show that A + AT is symmetric. Show that A+ ATY X. Ax = x X for…

A: Given  is a square matrix of order  .To show that matrix  is a symmetric matrix.And to show  for all…

Q: Suppose (B - C) D = 0, where B & C are m x n matricies and D is invertible. Show that B = C.

A: Given that  (B-C)D =0 here B and C are m×n matrices

Q: Suppose a, = (1 -1 1 1) 5. a, = (1 0 1 0)" a, = (1 1 1 1)'a, = (3 2 3 0)" · Please find out whether…

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Q: Exercise 3.1. Find f'(x), Vf(x) and V²f(x) for the following functions (a) f(x) = }r"Hx, where H is…

A: Given, fx=12xTHx, where H is an n×nconstant matrix.…

Q: Show that ||A||2 =  √(ρ(AA*))  where ρ(AA*) is the spectral radius of the AA* matrix.

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Q: Show that f(x): Rn -->R is a convex function f(x)=max{ ||APx-b|| , P is a Permutation matrix}…

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Q: Express the model Y₁ = Bo + Bixi + B₁r? + &i where the ɛ; have mean zero, variance o² and are…

A: Given Yi=β0+β1xi+β11xi2+εi ,i=1,2,......5

Q: 3. If A is a nonsingular matrix, then A' is nonsingular and its inverse is (A')1 = (A') '. Prove it!

A: A matrix A is said to be invertible if and only if  det(A) is not equal to zero i.e.  |A| ≠ 0

Q: (BH) Recall that a matrix Q is orthogonal if Q' Q = I.

A: Given, A matrix Q is orthogonal if QTQ=I Proof: (a) The one way to write the norm for vector x is…

Q: 2 - P=[1] and o-[11] 41 Q = Find a 2 x 2 matrix X such that PXQ= = -4 27

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