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Show that if A is a square matrix that satisfies the equation A2 - 2A + I = O, then A = 2I - A The equation A - 2A I = O implies that It follows that Notice ---Select--- ---Select--- the last equation means that multiplied with A is the identity, which is what we wanted to prove. -Select--
7. Matrix A is said to be involutory if A2 = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.
Problem 3. Give the definitions of an invertible square matrix and of the inverse of Let A be a square matrix. List at least five conditions that are equivalent to A being Prove that the inverse of a square matrix is unique if it exists. a square matrix. invertible. Problem 3. Give the definitions of an invertible square matrix and of the inverse of Let A be a square matrix. List at least five conditions that are equivalent to A...
2 0 If A is a square matrix then A2 = AA. Let A = Find A2 3 1 A2 = (Simplify your answer.)
(c) If A is a square matrix and A2 = 0,then A = 0. (d) Let A, B be two square matrices. If (A + B) 2 = A2 + 2AB + B2 , then AB = BA.
[3] 7. Let A be a square matrix such that A# I and A+ -I with eigenvalue X. Prove that if AP = I(I is the identity matrix), then = 1 or = -1.
. Let A be an n × n matrix. Prove that dim(span({In, A, A2,...})) ≤ n.
(3) 7. Let A be a square matrix such that A# I and A+ - with eigenvalue A. Prove that if AP = (is the identity matrix), then X = 1 or X = -1.
, then n lim Let Ά be a square matrix. Prove that if ρ(A)<1 Use the following fact without proof. For any square matrix A and any positive real number ε , there exists a natural matrix norm I l such that l-4 ll < ρ (d) +ε IIA" 11-0
Let A be an matrix. Prove that We were unable to transcribe this imagedim(span(In, A, A2....))< n dim(span(In, A, A2....))