3CGiven the Euclidean inner product space (R³(R),+,,*), where for each_x=(X₁,X2,X3),y=(yı,y2,y³)=R³,x*y = 2(x¹y₁+x³y3) + X2(Y1+Y3) + (x1+x3)y2 + 3x2y2.Show that matrix[2 1 01A = 1 3 1LO 1 2is invertible and calculate matrix A-¹ using the Cayley-Hamilton theorem.

Answered: Image /qna-images/answer/c8782680-86a0-43dd-8dbf-5f3d26c8882d.jpg

Answered: Given the Euclidean inner product space…

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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Does the set of all 2x2 matrices form a vector space, please explain.
Find orthogonal bases for the null space and for the range of the matrix [1, 0, 0, 3; 1, 1, 0, 4; 2, 0, 3, 8; 1, 1, 3, 6]
Answer only part c
Q) Let A be diagonalizable so that P-1AP = D. Show that A2 must be diagonalizable.
3. Let vi = [1,-2,3], v2 = [-1,–4,5], and V3 [5,-1,3] be vectors in R3. a) Use determinant to show that v1, v2, and v3 are linearly dependent. b) Use Gauss-Jordan elimination to write vị as linear combination of v2 and v3.
3. Find all matrices A such that: 936 3 5 4 6 4 6 b) AT A = 81₂ + A² c) (B-¹(AC))¹ = 5(C-¹B) where Band Care any 3x3 invertible matrices. a) ATA = and A is an upper triangular matrix with positive entries on the main diagonal. and A is a 2x2 skew-symmetric matrix.
= 1) Suppose that A² A and B is a diagonal matrix similar to A. Prove that B² = B and that the entries on the diagonal of B are only 0 and 1.
please help
Let y = 0 and u = Into A enter UTU. Into B enter UUT. Into Center projwy. Into D enter (UUT)y. T let U be the 2x1 matrix whose only column is u, and let W span(u).
While trying to invert A, [A I] is carried to [P Q] by row operations. Show that P=QA.
Compute the matrix products if possible
(2) Let aj = (1,2, -1), a2 = (3,0, 1) and bị = (-1, 10, 7), b2 = (1, 2, 0) in R³. %3D (a) Is a, in Span(a,)? (b) Which of bị, b2 is in Span(a, a2)? (c) Write each b; in Span(a1, a2) as a linear combination of a1, a2.

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Given the Euclidean inner product space (R³(R),+,,*), where for each x=(x1,x2,X3), y=(yı,y2,y3) ER³, x*y = 2(x₁y1+x3y3) + x2(y1+y3) + (x1+x3)y2 + 3x2y2. Show that matrix [2 1 0 A = 1 3 1 LO 1 2] is invertible and calculate matrix A¹ using the Cayley-Hamilton theorem.

Given the Euclidean inner product space (R³(R),+,,), where for each x=(x1,x2,X3), y=(yı,y2,y3) ER³, xy = 2(x₁y1+x3y3) + x2(y1+y3) + (x1+x3)y2 + 3x2y2. Show that matrix [2 1 0 A = 1 3 1 LO 1 2] is invertible and calculate matrix A¹ using the Cayley-Hamilton theorem.