Math final

If the augmented matrix corresponding to a system of three linear equations in three variables has a row of the form 0 0 0 /a ,where a is a nonzero number, then the system has no solution. True. The row with the given form says 0x + 0y + 0z + 0a = 0. But if a ≠ 0, we have a contradiction. True. The row with the given form says 0x + 0y + 0z = a, or 0 = a. But if a ≠ 0, we have a contradiction. False. The row with the given form says 0x + 0y + 0z + 0a = 0. There is no contradiction; instead there are an infinite number of solution. False. That a row has the given form indicates that the matrix is not yet in row-reduced form, and more row operations are needed to determine the answer. False. The row with the given form indicates that an error was made in the calculation; this form should not occur in a correctly reduced matrix.

2nd true

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The matrix A = a b c d has an inverse if and only if ad − bc = 0. True. This is from the Formula for the Inverse of a 2 × 2 Matrix. False. The matrix A has a multiplicative inverse if and only if ad − bc > 0. False. The matrix A has a multiplicative inverse if and only if ad − bc ≠ 0. False. The matrix A has a multiplicative inverse if and only if ab − cd < 0. False. The matrix A has the identity matrix as an inverse if ad − bc =

3rd down

You hear the following on your local radio station's business news: "The economy last year grew by 28%. This was the second year in a row in which the economy showed a 28% growth." This means that, in dollar terms, the economy grew more last year than the year before. Why? The economy last year was smaller than the year before; 28% of last year's economy is smaller than 28% of the year before. The economy last year was equal to the year before; 28% of last year's economy is the same as 28% of the year before. The economy last year was larger than the year before; 28% of last year's economy is larger than 28% of the year before.

3rd down

If A and B are square matrices with AB = I and BA = I, then which of the following is true? A and B must be equal. A and B must both be singular. At least one of A and B is singular. B is the inverse of A.

B is inverse of A

An investment that earns 11% (compound interest) every year is the same as an investment that earns 5.5% (compound interest) every 6 months. True False

False

While shopping for a car loan, you get the following offers: Solid Savings & Loan is willing to loan you $30,000 at 3% interest for 4 years. Fifth Federal Bank & Trust will loan you the $20,000 at 6% interest for 3 years. Both require monthly payments. You can afford to pay $650 per month. Which loan, if either, can you take? Solid Savings & Loan Fifth Federal Bank & Trust neither loan

Fifth fed and trust

Fleet Street Savings Bank pays interest at the rate of 4.25%/year compounded weekly in a savings account, whereas Washington Bank pays interest at the rate of 4.125%/year compounded daily. Which bank offers a better rate of interest? Fleet Street Savings Bank Washington Bank

Fleet street savings bank

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, do the following. (Use x, y, z, and w as your variables, each representing the columns in turn.) 1 0 0 0 1 0 1 1 0 −5 0 0 0 1 6 (a) Determine whether the system has a solution. The system has one solution. The system has infinitely many solutions. The system has no solution.

Many

Determine whether or not the given pair of matrices is an inverse pair. HINT [See Quick Examples page 207.] A = 1 1 4 0 1 2 0 0 1 B = 1 −1 0 0 1 −2 0 0 1

No

Determine whether or not the given pair of matrices is an inverse pair. HINT [See Quick Examples page 207.] A = 0 4 4 0 B = 0 4 4 0

No

Determine whether or not the given pair of matrices is an inverse pair. HINT [See Quick Examples page 207.] A = 4 0 0 3 B = 1/4 0 0 1/4

No

Indicate whether the matrix is in row-reduced form. 0 1 /2 0 0 /4 Yes, the matrix is in row-reduced form. No, the matrix is not in row-reduced form

No

Indicate whether the matrix is in row-reduced form. 1 0 0 −2 0 1 0 −8 0 0 2 −4 Yes, the matrix is in row-reduced form. No, the matrix is not in row-reduced form.

No

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, do the following. (Use x and y as your variables, each representing the columns in turn.) 1 0 9 0 1 4 0 0 0 (a) Determine whether the system has a solution. The system has one solution. The system has infinitely many solutions. The system has no solution.

One

You are offered three investments. The first promises to earn 17.5% compounded annually, the second will earn 17% compounded quarterly, and the third will earn 16.5% compounded monthly. Which is the best investment? 17.5% compounded annually 17% compounded quarterly 16.5% compounded monthly

Quarterly

A system of three equations in two unknowns corresponds to three lines in the plane. Describe several ways that these lines might be positioned if the system has no solutions. (Select all that apply.)

The lines are all parallel. The three lines intersect in three different points (forming a triangle). Two of the lines are parallel.

In the matrix of a system of linear equations, suppose that one of the rows is a multiple of another. What can you say about the row-reduced form of the matrix? The row-reduced matrix has two rows of zeros. The row-reduced matrix has two columns of zeros. The row-reduced matrix has a column of zeros. The row-reduced matrix has a row of ones. The row-reduced matrix has a row of zeros.

The row reduced matrix has a row of zeros

Select one: The addition or subtraction of two linear equations results in the equation 0 = 36. Then the graphs of those equations are parallel lines. Then the graphs of those equations are perpendicular lines. Then the graphs of those equations are the same line. Then the graphs of those equations are not parallel lines.

Then the graphs of those equations are parallel lines.

The addition or subtraction of two linear equations results in the equation 0 = 0. -the graphs of those equations are the same line. -Then the graphs of those equations are parallel lines. -Then the graphs of those equations are perpendicular lines. -Then the graphs of those equations are none of the above.

Then the graphs of those equations are the same line.

For what kind of compound interest investments is the effective rate greater than the nominal rate? -When the interest is compounded less often than once a year. -When the interest is compounded more than once a year. When is it smaller? -When the interest is compounded less often than once a year. -When the interest is compounded more than once a year.

When the interest is compounded more than once a year When the interest is compounded less often than once a year

Select one: Adding two linear equations gives x = 38 and subtracting them gives y = 38. Then the graphs of those equations are not parallel. Then the graphs of those equations are perpendicular. Then the graphs of those equations are parallel. Then the graphs of those equations are equal.

are not parallel

A system composed of two linear equations must have atleast one solution if the straight lines are represented by these equations are parallel

false

If the row reduced form of the augmented matrix corresponding to a system of linear equations has a row of zeros, then there are infinitely many solutions

false

Karen has been depositing $160 at the end of each month in a tax-free retirement account since she was 25. Matt, who is the same age as Karen, started depositing $240 at the end of each month in a tax-free retirement account when he was 35. Assuming that both accounts have been and will be earning interest at the rate of 7.5%/year compounded monthly, who will end up with the larger retirement account at the age of 65? Karen Matt

karen

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, do the following. (Use x, y, and z as your variables, each representing the columns in turn.) 1 0 0 4 0 1 0 9 0 0 0 0 (a) Determine whether the system has a solution. The system has one solution. The system has infinitely many solutions. The system has no solution.

many

If a linear programming problem has an unbounded, nonempty feasible region, then optimal solutions must be one of the following. They cannot exist. They must exist. They may or may not exist.

may or may not

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, do the following. (Use x, y, and z as your variables, each representing the columns in turn.) 1 0 0 6 0 1 0 −3 0 0 1 9 (a) Determine whether the system has a solution. The system has one solution. The system has infinitely many solutions. The system has no solution.

one

A system composed of two linear equations in two variables must have at least one solution if the lines represented by these equations are nonparallel. True False

true

Determine whether the statement is true or false. Choosing the pivot row by requiring that the ratio associated with that row be the smallest ensures that the iteration will not take us from a feasible point to a non-feasible point. True False

true

Determine whether the statement is true or false. If A is a matrix, then (AT)T = A. True False

true

Determine whether the statement is true or false. If, at any stage of an iteration of the simplex method, it is not possible to compute the ratios (division by zero) or the ratios are negative, then we can conclude that the standard linear programming problem has no solution. True False

true

Your friend Frans tells you that the system of linear equations you are solving cannot have a unique solution because the reduced matrix has a row of zeros. Comment on his claim. The claim is right. The claim is wrong.

wrong

The augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations. 0 1 0 1 2 0 0 1 −5 3 0 0 0 0 0 0 0 0 0 0 (a) Determine whether the system has a solution. Yes No

yes