## 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

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