**Problem 6.**(a) If \( A^2 = I \), what are the possible eigenvalues of \( A \)?(b) If this \( A \) is 2 by 2, and not \( I \) or \(-I\), find its trace and determinant.(c) If the first row is \( (3, -1) \), what is the second row?

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Answered: Problem 6. (a) If A² = I, what are the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Problem 2 5a – 7 -10a + 47 -10a + 41 Let M(a) = -a + 2 2a + 8 2a – 1 2а-4 -4a + 14 -4a + 17 (a) Find all values of a for which M(a) has some 2-dimensional eigenspace (for some eigenvalue). You might find it useful to know that 3 is an eigenvalue of M(a) for all values of a. (b) For each value that you find in (a), find a matrix P for which P- M(a)P = D, where D is diagonal. (c) Find all values of a for which M(a) is diagonalizable.
In Step 2 (Description of the steps), I see that you're finding the eigen values of matrix A. I see that you set A - (lambda)i = 0. Next you say "Solving this, we get (lambda2-1) (lambda-2)2 = 0 ---> lambda = 1, -1, 2, 2". Can you please explain how solved to get the "(lambda2-1) (lambda-2)2 = 0 ---> lambda = 1, -1, 2, 2" ? I thought that I was supposed to find the characteristic polynomials (the values from the diagonals - in this case, 9-lambda, -10-lambda, etc.), set them equal to zero (9-lambda=0, -10-lambda=0, etc.) solve for the lambda values (lambda=9, lambda=-10, etc.), and use those lambda values in the D matrix.
mont Question 12 2 0 0 Matrix A = -2 2 0only has an eigenvalue = 2. 15 2 True O False Type here to search 6.
Solve
2. If one of the eigenvalues of [A]nxn is zero, it implies (a)The solution to system of equations [A][X] = [C] is unique (b) The determinant of [A] is zero (c) The solution to system of equations [A][X]= [0] is trivial (d) The determinant of [A] is nonzero
In this problem, if you give decimal answers then give at least three digits of accuracy beyond the decimal. The matrix has the following complex eigenvalues (give your answer as a comma separated list of complex numbers; use "i" for ✓-1 and feel free to use a computer to solve the relevant quadratic equation): λ = 1.65+1.548386257i, 1.65-1.548386257i Since A has non-real eigenvalues, it is not diagonalizable, but we can find a matrix C = section 5.5 of Lay): C = P = cos(6) A = sin(8) -sin(8) cos(8) [113] 1.8 Further, we may factor Cas C = XY, where X is a matrix that scales by the positive real number radians, with —π/2 ≤0 ≤ π/2) is and an invertible matrix P such that A = PCP-¹ (I want you to cook up C, P as in and Y is rotation matrix whose counter-clockwise rotation angle (in This means that if we let B be the basis for R² consisting of the columns of P, then the B-matrix of A is C. So if we're willing to change our basis for IR², then the linear transformation x Ax really is just…

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Problem 6. (a) If A² = I, what are the possible eigenvalues of A? (b) If this A is 2 by 2, and not I or -I, find its trace and determinant. (c) If the first row is (3,-1), what is the second row?