(3) The exponential of a number x can be computed through the Taylor series e=1+2+2/2++"/n!+... e² = 1 + x + x² / 2 + ··· + x^ /n! + ... The exponential of a square matrix A is defined by the Taylor series: A² A³ 2 3! An e₁ = I + A+ + + + +... where I is the identity matrix of the same size as A. n! (a) Find e for 0 2 3 B == 004 000 As you will learn, the solution of the system of first order differential equations d x= Ax dt with initial condition (0) Fois = x(t) = e¹¹ão, where et=I+ At+ (At)² (At)³ (At)" + + + + 2! 3! n! A square matrix A is said to be diagonal, if all of its entries outside of the main diagonal, (a), are equal to zero. If A is a diagonal matrix, then it is very easy to calculate e, For example, let Then eA - (61) + ( 0 -(32) (1+3+ (3)² + .. -(63) 0 A 1-(2) + ( 3 0 02 2! 0 (2)2 + 3! +. 2! 0 3! 0 1+2+(2)²+ -1 (b) What is e, where D = 0 40 ? 0 00

(3) The exponential of a number x can be computed through the Taylor series e=1+2+2/2++"/n!+... e² = 1 + x + x² / 2 + ··· + x^ /n! + ... The exponential of a square matrix A is defined by the Taylor series: A² A³ 2 3! An e₁ = I + A+ + + + +... where I is the identity matrix of the same size as A. n! (a) Find e for 0 2 3 B == 004 000 As you will learn, the solution of the system of first order differential equations d x= Ax dt with initial condition (0) Fois = x(t) = e¹¹ão, where et=I+ At+ (At)² (At)³ (At)" + + + + 2! 3! n! A square matrix A is said to be diagonal, if all of its entries outside of the main diagonal, (a), are equal to zero. If A is a diagonal matrix, then it is very easy to calculate e, For example, let Then eA - (61) + ( 0 -(32) (1+3+ (3)² + .. -(63) 0 A 1-(2) + ( 3 0 02 2! 0 (2)2 + 3! +. 2! 0 3! 0 1+2+(2)²+ -1 (b) What is e, where D = 0 40 ? 0 00

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 67E

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