Let A= .  1. Find all the eigenvalues of A.   (12 points) 2. Find the corresponding eigen vectors.   (12 points) 3. Is A nonsingular, that is, does A have an inverse? Answer this just using the knowledge of the eigen values. Give reasons for your answer. (6 points)  

Consider the system of linear equations: x_1+3x_2+x_3+x_4=3 2x_1-2x_2+ x_3+2x_4=8 3x_1+ x_2+2x_3 -x_4=-1. I.  (a) Write the corresponding augmented matrix.  (b) Do suitable row operations so that first column is all zero except for the  entry in the first row.       (3 points each).   II. (a) Complete computing an echelon form for the above matrix.  (8 points)       (b) Find all possible solutions .   (4 points) (c) How many solutions are there? Which are the basic variables and which are the free variables, if any? (2 points) III. Given the matrix A= ,  find A^{-1}. (10 points)

I.  (i) Given vectors v_1= , and v_2 =,  determine their span as a plane in 3-space.  (10 points)  (ii) Is the vector x =  in the span of v_1 and v_2? Give reasons for your answer. (5 points) II.  Given the matrix A= ,  (i) find an echelon form for it. (5 points)     (ii) Determine the null-space of A, using the echelon form in (i).  (5 points)     (iii) Determine a basis for the null-space of A. (5 points)

I.  (i) Given vectors v_1= , and v_2 =,  determine their span as a plane in 3-space.  (10 points)  (ii) Is the vector x = in the span of v_1 and v_2? Give reasons for your answer. (5 points) II.  Given the matrix A= ,  (i) find an echelon form for it. (5 points)     (ii) Determine the null-space of A, using the echelon form in (i).  (5 points)     (iii) Determine a basis for the null-space of A. (5 points)

  Let A= .  1. Find all the eigenvalues of A.   (12 points) 2. Find the corresponding eigen vectors.   (12 points) 3. Is A nonsingular, that is, does A have an inverse? Answer this just using the knowledge of the eigen values. Give reasons for your answer. (6 points)        

Please answer questions 1 and 2 on page 1 and questions 3 and 4 on page 2. Let A= .  1. Find all the eigenvalues of A.   (7 points) 2. Find the corresponding eigen vectors.   (7 points) 3. Can A be diagonalized?   (2 points) 4. If the answer to 3 is yes, find a matrix X and a diagonal matrix D so that X^{-1}*A*X is a diagonal matrix D. (4 points) Old Quiz 6: ————- 1.  Use Cramer’s rule to find the solutions of the system of equations: x1+ 3×2+ x3=1 2×1+x2+x3  =5 -2×1+2×2 – x3= -8   This can broken into the following subproblems: (i) Compute the determinant of the coefficient matrix for the system of equations above.  (5 points) (ii) Compute the determinant of the three matrices obtained by replacing one column of the coefficient matrix by the vector on the right hand side.  (4 points each) (iii) Write the solutions for each of the variables x1, x2 and x3. (3 points)    

Answer question 1 on page 1 and question 2 on page 2. Let v= and w=. 1.  Find an orthonormal basis for the Span of the vectors v and w. (12 points: 4 points for the first vector and 8 points for the second vector) 2. Let b =. Find the orthogonal projection of the vector b in the Span of v and w. (8 points)      

Penguins that live in the South Pole must deal with the freezing temperature and cold winds. What type of ecological factor is this?

Which is not a characteristic of the phylum Platyhelminthes (flatworms)?

What is Müllerian mimicry?