3. Let A be an n x n positive semidefinite matrix. Let B be an nx n positive definite matrix. Then we have Klein's inequality tr(A(ln(A) – In(B))) > tr(A – B). (i) Let 1/2 -1/2) -i/2 1/2 (1/2 B = (O 1/2). A = Calculate the left-hand side and the right-hand side of the inequality. (ii) When is the inequality an equality?

Q: 1. Give an example of a 2 x 2 matrix with no inverse ( singular). Show how it become singular 2.…

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Q: 1 -2 -1 3 Let A = 4 -3 5 2 1 Let ez be the third column of the 4 x 4 identity matrix I4. Compute Aez…

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Q: (b) Compute the matrix exponential function exp(-A) where [1 A = 2

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Q: (a) Show that A2 − 2A + 5I = O, where I is the identity matrix of order 2.(b) Show that A−1 = 1/5…

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Q: Find a matrix A such that [is: is a solution for AX=X'.

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Q: 8. Let X be an n x p matrix of full column rank. (a) Use Theorem A.23 to show that (In + XX')- = In…

A: X be an n × p matrix.(a) Let A=In, B=X, C=Ip, D=X'Then (A+BCD)-1=(In+X Ip,X'-1=(In+XX')-1 and…

Q: 2. If the product of matrices cos 0 A = cos O sin 0 cos e sin 0 sin² 0 cos' o B = cos o sin cos o…

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Q: Given a square matrix 3 x 3 such that 2AA' = A' and Det(A) + 0, find Det(4). %3D

A: Let the square matrix of 3×3 is A s.t

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Q: Let B = {(1, 3), (-2, -2)} and B' = {(-12, 0), (-4, 4)} be bases for R², and let [:] 0 3 A = be the…

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Q: Suppose A-A, where A is the matrix. The det (A) is

A: Determinant of a null matrix is equal to zero.

Q: 5. Let A be a (3×3)-matrix with the characteristic polynomial pA(x) = (2 – x)(x² + 1– a²) for some a…

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Q: 1. If 1+ A is nonsingular, show that (I+ A)"' and I– A commute and (I + A) +(I+ A')"=1, where I is…

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Q: A X 2) Consider a matrix M = (6 %) in block form, where A and B are square matrices. Prove that 0 B…

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Q: If B is invertible matrix, then (5BT) = (A) 금BT (B) 5 (B-1)" (C) 5 (B")- (D) B-1 (E) (B-1)"

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Q: Let A be the matrix 5 3 2 Find p(A). p(x) = x³ – 2x + 4 p(A) =

A: Given: A=5312 and p(x)=x3-2x+4.