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6. [²3 Find the eigenvalues and eigenvectors of A =

6. [²3 Find the eigenvalues and eigenvectors of A =

Advanced Engineering Mathematics
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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6. 7. 8. =[2²3] Find the eigenvalues and eigenvectors of A = 3 1 0 For A = 0 1 0 find a diagonal matrix and an invertible matrix S so that 421 A=SAS¹. Then use these results to find 4100 Solve_y"+5y'+6y=0 with y(0)=0 and y'(0)=1
Transcribed Image Text:6. 7. 8. =[2²3] Find the eigenvalues and eigenvectors of A = 3 1 0 For A = 0 1 0 find a diagonal matrix and an invertible matrix S so that 421 A=SAS¹. Then use these results to find 4100 Solve_y"+5y'+6y=0 with y(0)=0 and y'(0)=1
1. Given the matrix A below with Det A = 4, find the determinants of B and C for the matrices below. 2. Prove: a. b. C. 3. 4. az az A = b₁ b₂ G C₂ 5. az b3 C₂ az az B = b₂ b₂ C3 C₂ a₁ b₁ C₁ a₂ a₁ C= _b₂ +4q₁ by +4C₂ C2 9₂ If A is nonsingular and A² = A, then Det A = 1 If A = 0 (the zero matrix) for some n, then A is singular If A is 3 by 3 and skew symmetric, then Det A = 0 Find the volume of the box with one vertex at the origin and adjacent vertices at A (1, 4, 0), B ( -2, -5, 2) and C ( -1, 2, -1). Consider the unit sphere U: x² + y²+z² = 1 What is its volume? Az bz + 4c3 What matrix A changes U to the ellipsoid E: +²+ = 1? Find the volume of this ellipsoid using determinants. Use cross products to find a vector n normal to the triangle with vertices A, B, C in Problem 3 and whose length equals the area of this triangle.
Transcribed Image Text:1. Given the matrix A below with Det A = 4, find the determinants of B and C for the matrices below. 2. Prove: a. b. C. 3. 4. az az A = b₁ b₂ G C₂ 5. az b3 C₂ az az B = b₂ b₂ C3 C₂ a₁ b₁ C₁ a₂ a₁ C= _b₂ +4q₁ by +4C₂ C2 9₂ If A is nonsingular and A² = A, then Det A = 1 If A = 0 (the zero matrix) for some n, then A is singular If A is 3 by 3 and skew symmetric, then Det A = 0 Find the volume of the box with one vertex at the origin and adjacent vertices at A (1, 4, 0), B ( -2, -5, 2) and C ( -1, 2, -1). Consider the unit sphere U: x² + y²+z² = 1 What is its volume? Az bz + 4c3 What matrix A changes U to the ellipsoid E: +²+ = 1? Find the volume of this ellipsoid using determinants. Use cross products to find a vector n normal to the triangle with vertices A, B, C in Problem 3 and whose length equals the area of this triangle.
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