تیا مصر مهمة جگہ Page 47 (Lecture 10) Theorem: IP D. is a fundamental matrix of homo- geneous system X° = AX_then_x(t) = (1)_f_&_(s)_g(s)_ds). to is a solution of non homogeneous system_x = Ax+g(t). Condition x(t)=0_and_X-(+) = O(+)-(0)-X-+ _with t --(+) S² (= (s)_g_(s)_ds with condition_X(6) = x. to اله ها How CamScanner

:} -2 1 2 2 1 (a) Write as a linear combination of the set of matrices -1 3 Do this "directly" without using a coordinate mapping. That is, solve 1 1 11 -2 +b which will lead to a system of equations. %3D -1 1 2 1 y (b) Show that the set does not span V = :x, y, z eR} by 5 3 finding a vector (a matrix in this case) that is in V but is not in 1 span 1 Please make it clear that (or show why) your vector satisfies 0 5 3. the given conditions. Hint: You may use a coordinate mapping, but you need to define your basis and write clear explanations of your findings. 35 -11 35 1 (c) Let b, 1 -11 b, and w= Write %3D as a 5 3 78 78 1 linear combination of 2 and 5 2 1 Solve this by using a 3

Write the equivalent first-order system in the form of x' = Ax. Find the Coeffi- cient matrix. Also determine the system is liner or non-linear and homogeneous and nonhomogeneous "1 (a) x + x - x = 0 (b) x "1 3x + 4x = 2-t Please help with detailed answers for part a and b.

True or False? Justify your answer. a) Let A, B, and C be two square matrices of the same dimension and A be an invertible matrix then ((A^−1)BCA)^2023 = A^(−1)xB^(2023)xC^(2023)xA. b) Let A be an n×n matrix. The set of all n×n matrices X that satisfies (A^2 −3I)X = O is not be closed under the matrix addition. (O denotes the zero matrix of the proper dimensions.) c) If A and B are 6×6 matrices of rank 6, then AB also has rank 6.

Let A = (2 1, 2 3) . Find a 2 x2 matrix X so that X^2 = A.

if the chractrestic polynomial of a matrix A is p(^) = (^ +1)(1 –2)² then det(-A) +tr (A) = 1 7 -2 none 2 3.

Explain this.kindly correctly and handwritten

Please show all steps

find a fundamental system for the linear system of the first order. -1 j'(t) = -5 -17 a Note:please write your solution e(A-t)in the form: [a*e^(lambda*t),b*e^(lambda*t)] ý;(t)

For matrix (0 -1 0, 1 0 0, 0 0 1), show that [adj(A)}^-1 = (1 / det(A))*A = adj (A^-1)