Let A = \begin{pmatrix}1&2&0\\ 0&1&-2\\ 2&2&-1\end{pmatrix}. For the determinant (λ-1)^(2)(λ+1) = 0. So, the eigen value λ = 1 with algebraic multiplicity of 2, and = -1 with algebraic multiplicity of 1. For λ = 1: Use Av = λv: \begin{pmatrix}1&2&0\\ 0&1&-2\\ 2&2&-1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=1\begin{pmatrix}x\\ y\\ z\end{pmatrix}. For λ = -1 : Use Av = λv: \begin{pmatrix}1&2&0\\ 0&1&-2\\ 2&2&-1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=-1\begin{pmatrix}x\\ y\\ z\end{pmatrix}. Then Find P, P^(-1), D and then e^(tA) = P e^(tD) P^(-1)

Only (a) and (d) and (e).

2. Find a CR-decomposition for each complex matrix A and state the rank of A. -1 1 2+ i ^=671²1 i (a) A = 3 (b) A = 10i 6 - 2i -15i -9 -6+3i

Suppose A, B are 3x3 matrices, det A = 6 , and det B=-5. Find the following determinants. (a): det (B-'A) (b): det (AT B) (c): det (3B)

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 8x₁ + 3x₂ + 8x3, x'₂ = 3x₁ + 13x₂ + 3x3, x'3 = 8x₁ + 3x₂ + 8x3 What is the general solution in matrix form? x(t)

x'(t) = of the any - 2 5-6 4 1-4-6|x (t) x (0) = -2 + -3 1 The only eigen value of this matrix is -3, a triple root. You must explicitly find matrix involved, with the exception cannot involve Also, your answer 0 imaginary any matrix inverses. number i.

Consider the quadratic form Q(x, y) = -² A = la cla 14 9 ala 10 Ila 28 9 ---/- 14 9 14 9 14 9 20 9 56 x y 9 following principal minors of the matrix A: The (leading) principal minor 1pm3 = pm3 of order 3: The leading principal minor 1pm² = pm3 of order 2: The leading principal minor 1pm₁ = pm of order 1: The principal minor pm² of order 2: The principal minor pm² of order 2: The principal minor pm of order 1: The principal minor pm of order 1: 9 Now use your answers to classify Q: (No answer given) -28 x z We want to classify this using Sylvester's Criteria (the principal minor test). Calculate the 28 y z 9 + 202² with symmetric coefficient matrix