Q1 Let A be the 2 x 2 matrix a11 a21 a12 a22 where a11, a12, a21, a22 € R. Show that if a12@21 > 0, then A has two distinct real eigenvalues.

Q: V₁ = [1 0 0], V2 = [1 1 0], V3 = [011] are eigenvectors of A corresponding to A₁, A2, and A3,…

A:

Q: Let A be a 2 x 2 matrix with real coefficients and eigenvalues A1 = 3, and A2 = 5, which has…

A: Proceed as shown

Q: Explain how to determine whether an n x n matrix A is diagonalizable using... (a) similar matrices…

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Q: Let 1 be an eigenvalue of the n x n matrix A. Let B = A- AI. Show that B is not an invertible…

A: Let A be an n×n matrix. The characteristic polynomial of A is the function f(λ) given by…

Q: A be an orthogonal n by n real matrix where n is an odd number. Show that 1 or -1 must be an…

A:

Q: Let A be the 2 x 2 matrix a11 a12 a21 a22 where a11, a12, a21, 922 ER. Show that if a12021 > 0, then…

A: Consider the 2×2 matrix A=a11a12a21a22. Here a11, a12, a21, a22∈ℝ. Assume that a12a21>0. To show…

Q: Let A be a 3 x 3 matrix with eigenvalues 7/2, 7//4, 7/8. If the eigenvalues of A- are A1,2, 13, then…

A: we have given eigen values of matrix A these are 7/2,7/4,/7/8 As we know eigen value of A-1 =1/lemda…

Q: Let A be an n × n matrix and let B = A − αI for some scalar α. How do the eigenvalues of A and B…

A: To check that (eigenvalue of A- alpha) is an eigenvalue of B.

Q: Suppose that A is a matrix such that A2=0. Show that every eigenvalue of A is equal to 0.

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Q: Let A be a Hermitian matrix. Show that all its eigenvalues ​​belong to the real axis.

A: See proof below

Q: The characteristic polynomial of the matrix A = 4 -1 1 -1 -1 4 -1 4 -1 is (A-2)(X - 5)². a) Find the…

A: A matrix A is diagonalizable if there exists an invertible matrix P such that A=PDP-1 .

Q: Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are X₁ = 3, A₂ = −2, and A3 = -1 with…

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Q: 35. Give an example of a 4 x 4 matrix A without real eigen- values. ANS: Го -1 07 1 A = 0 0 -1 0 1

A: The answer is given as follows.

Q: Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is NOT diagonalizable. Show…

A: Consider the matrix 21202100-3 Since the given matrix is in upper triangular form , so the eigen…