Linear Algebra Matrix Practice Problems PDF

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LINEAR ALGEBRA MATH - 08...

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Exercises for Section 2. 1.

Let x

y

,

z

Calculate x + y and 3x− 5 y + z. 2.

Let

, x

y= -2, 2, 0, 4

Compute Ax, Ay, Ax + Ay, and A(x + y). 3.

Let .

Calculate each of the following quantities if it is defined: A + 3 B, A + C, C + 2D, AB,BA,CD,DC . 4.

Suppose A is a 2 × 2 matrix such that .

Find A. 5.

Let ei denote the n × 1 column vector, with all entries zero except the ith which is 1, e.g., for n = 3, e

,

e

,

e

.

Let A be an arbitrary m × n matrix. Show that Aei is the ith column of A. You may verify this just in the case n = 3 and A is 3×3. That is sufficiently general to understand the general argument. 6.

Write each of the following systems in matrix form. (a) 2x1 − 3x2 = 2 −4x1 + 2x2 = 3 (b) 2x1 − 3x2 = 4 −4x1 + 2x2 = 1 (c) x1 + x2 = 1 x2 + x3 = 1 2x1 + 3x2 − x3 = 0

7.

(a) Determine the lengths of the following column vectors

(b) Are any of these vectors mutually perpendicular? (c) Find unit vectors proportional to each of these vectors. 8.

One kind of magic square is a square array of numbers such that the sum of every row and the sum of every column is the same number. (a) Which of the following matrices present magic squares?

(b) square. 9.

Use matrix multiplication to describe the condition that an n × n matrix A presents a magic

Population is often described by a first order differential equation of the form

where p represents

the population and r is a parameter called the growth rate. However, real populations are more complicated. For example, human populations come in different ages with different fertility. Matrices are used to create more realistic population models. Here is an example of how that might be done Assume a human population is divided into 10 age groups between 0 and 99. Let xi, i = 1,2,...,10 be the number of women in the ith age group, and consider the vector x with those components. (For the sake of this exercise, we ignore men.) Suppose the following table gives the birth and death rates for each age group in each ten year period. i Age BR DR 1 0...9 0 .01 2 10...19 .01 .01 3 20...29 .04 .01 4 30...39 .03 .01 5 40...49 .01 .02 6 50...59 .001 .03 7 60...69 0 .04 8 70...79 0 .10 9 80...89 0 .30 10 90...99 0 1.00 For example, the fourth age group is women age 30 to 39. In a ten year period, we expect this group to give birth to .03x4 girls, all of whom will be in the first age group at the beginning of the next ten year period. We also expect .01x4 of them to die, which tells us something about the value of x5 at the beginning of the next ten year period. Construct a 10 × 10 matrix A which incorporates this information about birth and death rates so that Ax gives the population vector after one ten year period has elapsed. Note that Anx keeps track of the population structure after n ten year periods have elapsed....