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Semester Summer Final Examination, 2020

MATH1051 Calculus & Linear Algebra 1

PART A 20 marks

● Part A is Pass/Fail. A student who Passes Part A earns all 20 marks. A student who Fails Part A earns 0 marks.

● The Pass threshold is 13/20.

● There are 20 questions (each question is worth one mark) on: Gaussian elimination, Inverses, Determinants, Vector Products, Eigenvalues, and Eigenvectors.

1) The augmented matrix of a linear system has been reduced by row operations to the form

How many solutions does this system have?

2) Find the general solution to the following system of equations:

3) Apply elementary row operations until the augmented matrix

is in Gauss reduced form.

4) Determine the nullspace of the following matrix:

5) Apply elementary row operations until the augmented matrix

is in Gauss reduced form.

6) Let A and B be invertible matrices. Is the following statement True?

(AB)⁻¹ = A⁻¹B⁻¹.

7) Let A be a square matrix.

Give three different conditions equivalent to the statement “A is invertible.”

8) Let A be an invertible matrix, and let b be a vector. How many solutions are there to the equation Ax = b?

9) Determine the inverse of the following matrix:

10) Find the determinant of the following matrix:

11) Suppose A and B are matrices, with |A| = 2, and |B| = 5.

Evaluate |AB|.

12) A matrix C has determinant 3.

Which of the following is true? Circle the correct statement:

(a) C is invertible.

(b) C is not invertible.

(c) C may be invertible., but there is not enough information to determine for sure.

13) Given that 

14) Let A be a matrix, and suppose |A| = 3.

Evaluate |A⁻¹|.

15) Let u = (0, 1, 1), and v = (1, 0, 3).

Calculate u × v.

16) Suppose u × v = 5j.

Determine (−3u) × v.

17) Let a and b be non-zero vectors, and suppose a × b = 0.

Which of the following is true? Circle the correct statement:

(a) a and b are parallel.

(b) a and b are perpendicular.

(c) a and b may be either parallel or perpendicular, but there is not enough information to determine for sure.

18) Find all eigenvalues for the following matrix:

19) One of the eigenvalues of matrix

Find an eigenvector corresponding to this eigenvalue.

20) One of the eigenvectors of matrix 

What is this eigenvector’s corresponding eigenvalue?

PART B - CALCULUS - 40 marks

21) Determine whether the following series converge. Show all working.

22) Recall the double angle formula:

Find the MacLaurin series for f(x) = sin² x. Show all working.

23) a) Show that 

b) Evaluate the volume of the solid obtained by rotating  about the x-axis over the interval [0, 1].

24) a) Explain why we would use rule to determine 

b) Evaluate  (if the limit exists).

PART B - LINEAR ALGEBRA - 30 marks

25) a) Let G be a square matrix such that GG^T = I.

Show that |G| = ±1 .

b) A matrix C is called idempotent if C² = C.

Let A be an m × n matrix and B an n × m matrix.

Prove that if BA = I, then AB is idempotent.

c) Let F be a square matrix. Show that if 3 is an eigenvalue of F, then 9 is an eigenvalue of F².

26) a) Show that the following set of vectors is not linearly independent:

b) Do the following vectors form a basis for R³?

27) a) It is known that  is a vector space.

Find a basis for V. What is dim(V)?

b) Let

Show that W is not a vector space.