Description

Math1013, Assignment 3 2
Question 1. Inverse of a Matrix
5 pts
(a) Use row operations (show the sequence of steps), to compute the inverse of the matrix
A =⎡⎣
1 0 3
0 2 −1
−1 0 −2
⎤⎦
.
3 subpts
(b) Use A−1 to solve the system Ax = b where b =⎡⎣
1
−1
3
⎤⎦
. 2 subpts
Question 2. Matrix algebra
5 pts
Let A be a 2 × 2 matrix such that A% 1
−1& = % 2
−3& and A%2
2& = %0
2&.
(a) Use this information to write 2 × 2 matrices V and W such that AV = W.
(Hint: No calculations are needed to do this.) 2 subpts
(b) Use matrix algebra (including inverses) to calculate A. 3 subpts
Question 3. Working with Inverses
5 pts
(a) Let
A =⎡⎣
0 0 0
1 0 0
0 1 0
⎤⎦
.
(i) Show that A3 = 0. 1 subpts
(ii) Find (I − A) and (I + A + A2) and compute the product (I − A)(I + A + A2).
Hence state the inverse of (I − A). 2 subpts
(b) Suppose A is a square matrix (of unspecified size) such that A4 = 0. From observations
in part b), write an expression for the matrix (I −A)−1 and use matrix algebra
to confirm that (I − A)(I − A)−1 = I. 2 subpts
Math1013, Assignment 3 3
Question 4. Midpoint Rule
8 pts
You will need a calculator for this question.
(a) Approximate the integral ‘1
0 e−x
2
dx by using the midpoint rule with two subintervals
(i.e. n = 2). Calculate the maximum error and include an error bound with your
answer.
6 subpts
(b) How many subintervals would be required to achieve an approximation correct to
four decimal places (i.e. an error less than .00005)? 2 subpts
Question 5. Volumes
7 pts
(a) The base of a solid is a diamond with vertices located at (1, 0), (0, 2), (−1, 0), (0,−2).
Each cross-section perpendicular to the y-axis is a semicircle.
Set up an integral and use it to find the volume of the solid. 5 subpts
(b) Show that by cutting the solid of part (a), we can rearrange the pieces to form a cone.
Use the formula for the volume of a cone to check your answer to part (a). 2 subpts