Suppose A is a 3 x 3 matrix with eigenvectors [1] V1 = V2 = 1 and v3 = corresponding to eigenvalues A1 =-, d2 = }, and A3 = 1, respectively. Find A and A20.

Find a 3 x 3 matrix • The vectors V₁ = satisfying the following properties: • A has two eigenvalues: A₁ = -3 and X₂ 3] 3 -3 and V₂ H -2 are eigenvectors of A corresponding to X₁. • The vector V3 = How to enter matrices. = then you should do it as follows: A = = -3. Enter the matrix A: a11 a12 a13 a21 a22 a23 a31 a32 a33 2 -3 is an eigenvector of A corresponding to A2. Martices should be entered row by row, enclosing each row in square brackets. There must be additional square brackets at the beginning and at the end of the whole matrix. For example, if you want to enter the matrix 2 [8 MIN HIN Do not forget about commas between matrix entries and between rows. A 2 [[2, -3/2, 4], [0, 1/2, 2]]

Suppose a 3 x 3 real matrix A has only two (real) distinct eigenvalues. Suppose that tr(A) = -1 and det(A) = 112. Find the eigenvalues of A with their algebraic multiplicities. smaller eigenvalue = has multiplicity and larger eigenvalue = has multiplicity

Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are X₁ = 3, A₂ = −2, and A3 = -1 with corresponding eigenvectors @· --D 0 1 0 Express A as PDP-1 where D is a diagonal matrix and use this to find A5. V₁ = A5 = V3 = 1, respectively.

Find the eigenvalues and eigenvectors for the coefficient matrix.

Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is NOT diagonalizable. Show WHY it is not diagonalizable.

Find eigenvector for 2 = 3 and given matrix A. X1 X2= X3= Submit the Answer 2 Question 2 grade: 5 3 | 10 A = [-5 17 10 -11 -21

the matrix A= }] has eigenvalues A₁ = 1 and A₂ = 5. The bases of the eigenspaces are v₁ = -12 Find an invertible matrix S and a diagonal matrix D such that S-¹AS = D. S= D= and v₂ = , respectively

Let A be a arbitrary n x n matrix, Show that if A1 and A2 are two eigenvalues with A1 # ^2 and eigenvectors vi and v2, respectively. Show that vị and v2 are linerly independent.

Find a 2x2 matrix A that has two eigenvalues 4, -1 whose eigenvectors 2 and 1 are