Math-260 (A. Farhat) Major Quiz No. 1 (003)

On Sec. 1.1-2.4

NAME:- St. No.:- July 2, 2001

Instructions:

v Answer all questions. Show all your work.

v Use a new page for each question.

v Submit your answers in the order of the questions.

1. Let and . If , find a, b, c, and d.

2. Let be a symmetric matrix. Find a, b, and c and then compute

3. Let A be a 4x4 symmetric, strictly upper triangular matrix. Find A.

4. Find the angle between the two vectors and .

5. Given the vector

a) Find a vector v that is perpendicular to u.

b) Verify your answer in part (a)

6. Without solving, state if the following systems of equations have (1) no solution, (2) one solution, (3) infinite number of solutions. Give a graphical representation of each system.

a) b)

c) d)

7. Solve the system by computing the row reduced echelon form of the augmented matrix.

8. Let the following matrices represent the row reduced echelon forms of the augmented matrices of linear systems. State whither the system has zero, one, or infinitely many solutions. Give the rank of each matrix.

a) b)

c) d)

9. Given the linear system

a) Determine t so that the system is consistent

b) Determine t so that the system is inconsistent

10. Answer the following questions by (T)rue or (F)alse.

________ 1) An matrix A is row equivalent to In if and only if the homogenous system Ax= 0 has only the trivial solution.

________ 2) For any vector , the inner product is always positive.

________ 3) If a matrix A is in reduced row echelon form, then it is possible for an element of a column with a pivot to be nonzero.