Exercise 2.8. Let a, b be real numbers and leta^= (₁ 1)A-61).What is AB ? Compute A² and A³. What is Am for an arbitrary integer m, and howto prove it?and B

Answered: Exercise 2.8. Let a, b be real numbers…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1 2 5 Let A = _^-[; ;] -[-]--[-]»-[?]-.-;] b₁ b3 7 16 29 12 a. Find A¹ and use it solve the four equations Ax=b₁, Ax=b₂ Ax=b3, and Ax=b4- a. Find A -5 1 31 b₂ = b. The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix [A b₁ b₂ b3 b4]. OA. The inverse matrix is A and b4= = Select the correct choice below and, if necessary, fill in the answer box to complete your choice 6 (Type an integer or simplified fraction for each matrix element.) O B. The matrix is not invertible. 34
Hint: What can you say about (?^2) + ??
Prove or give a counterexample: For any n greater than or equal to 1 and A, B in R^(n x n), one has det(A + B) = det(A) + det(B).
12:41 2) How many roots does the polynomial have ? a) 2x' - 4x? + 9x - 1 b) 2x - 9x - 1 1 3 4 5
a) z^2=w b)z^3=w c)z^10=w d)z^50=w e)z^6000=w
Prove that (-1+i3) + (-1 - if3)" has either the value 271 er the value -2" if n is ary n+1 Interer (positive Zero).
Problem 3. Let a, b and n be positive integers. Prove that (a) gcd(a", br) = gcd(a, b)” and lcm(a^, b") = lcm(a, b)”; (b) gcd(a-n, b.n) = gcd(a, b) · n and lcm(an, b⋅ n) = lcm(a, b) . n;

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Exercise 2.8. Let a, b be real numbers and let a ^= (₁ 1) A and B -(1) What is AB ? Compute A² and A³. What is Am for an arbitrary integer m, and how to prove it?