Applied Mathematics AEM 303 Quiz
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Questions and Answers

Linear algebra can be applied only to systems of linear equations.

False

Many exponential and power functions are readily convertible to linear functions and can be handled by linear algebra.

True

Rows are the numbers in a vertical line of a matrix.

False

Columns are the numbers in a horizontal line of a matrix.

<p>False</p> Signup and view all the answers

A square matrix has the number of rows equal to the number of columns.

<p>True</p> Signup and view all the answers

A column vector is a matrix composed of a single row.

<p>False</p> Signup and view all the answers

The dimensions of a column vector are given as r x 1.

<p>True</p> Signup and view all the answers

A row vector is a matrix having a single column.

<p>False</p> Signup and view all the answers

The dimensions of a row vector are given as 1 x c.

<p>True</p> Signup and view all the answers

When two matrices (A and B) are conformable for multiplication, the product (AB, i.e., matrix C) is said to be undefined.

<p>False</p> Signup and view all the answers

A null matrix is a matrix composed of all 0s and can be of any dimension.

<p>True</p> Signup and view all the answers

A coefficient matrix (A) is the matrix whose elements are the variables in a system of (linear) equations.

<p>False</p> Signup and view all the answers

A solution vector (X) is the matrix whose elements are the coefficients in a system of (linear) equations.

<p>False</p> Signup and view all the answers

A constant terms vector (B) is the matrix whose elements are the constants in a system of (linear) equations.

<p>True</p> Signup and view all the answers

A determinant is a single number or scalar and is found only for square matrices.

<p>True</p> Signup and view all the answers

A vanishing determinant is one expressed as |A| ≠ 0.

<p>False</p> Signup and view all the answers

A non-singular matrix is one whose determinant is expressed as |A| = 0.

<p>False</p> Signup and view all the answers

If ρ(A) < n, A is singular, and there is linear dependence.

<p>True</p> Signup and view all the answers

A minor matrix is a determinant with a prescribed sign.

<p>False</p> Signup and view all the answers

A triangular matrix is a matrix with zero elements everywhere above or below the principal diagonal.

<p>True</p> Signup and view all the answers

An adjoint matrix is the transpose of a minor matrix.

<p>False</p> Signup and view all the answers

The determinant of a matrix equals the determinant of its transpose: |A| = |A'|.

<p>True</p> Signup and view all the answers

If all the elements of any row or column are zero; the determinant is one.

<p>False</p> Signup and view all the answers

If two rows or columns are linearly dependent, the determinant is zero.

<p>True</p> Signup and view all the answers

A symmetric matrix is a matrix which when transposed equals the original matrix.

<p>True</p> Signup and view all the answers

In matrix algebra, an identity matrix is similar to number 1 in ordinary algebra.

<p>True</p> Signup and view all the answers

The expression $P(1 + \frac{r}{100})^n$ can be used to determine how money in a savings account grows over of time, where P is the original sum invested (principal), r is the interest rate, and n is the number of years (money is saved).

<p>True</p> Signup and view all the answers

Ordinarily, a quantity demanded is defined as a function of price.

<p>True</p> Signup and view all the answers

The economic convention is to plot price on the x-axis when plotting the demand curve.

<p>False</p> Signup and view all the answers

In economics, the inverse function is used when presenting the demand function.

<p>True</p> Signup and view all the answers

If y changes at a constant rate with respect to x, then the function relating y to x must be linear and its graph is a straight line.

<p>True</p> Signup and view all the answers

If a number a lies to the right of b on a number line, then it can be said that the number a is greater than the number b, written as a < b.

<p>False</p> Signup and view all the answers

If a number a lies to the left of b on a number line, then it can be said that the number a is less than the number b, written as a < b.

<p>True</p> Signup and view all the answers

In reality, most economic variables do not have linear relationships but non-linear relationships.

<p>True</p> Signup and view all the answers

Economic phenomena cannot be described with straight line curves but curves that are curvilinear in nature.

<p>True</p> Signup and view all the answers

The simplest non-linear function is known as a quadratic function and takes the form f(x) = ax2 + bx + c.

<p>True</p> Signup and view all the answers

Even if an economic function (e.g., demand function) is linear, functions derived from it (such as total revenue and profit) might be non-linear.

<p>True</p> Signup and view all the answers

The concept of the margin in economics is important because economists are usually interested in changes in economic variables.

<p>True</p> Signup and view all the answers

The concept of margin is important in economics, and the common marginal variables of interest are: marginal product, marginal cost, marginal revenue, and marginal propensity to consume/save.

<p>True</p> Signup and view all the answers

A retailer sells (poultry) eggs for ₦24. If a customer orders more than 100 eggs, the retailer is prepared to reduce the unit price by 4kobo on eggs bought above 100 pieces but up to a maximum of 300 pieces in a single order. The unit cost of the remaining 30 eggs is 24 − (0.04 × 30) = ₦22.80.

<p>True</p> Signup and view all the answers

If the total cost of the eggs bought in excess of 100 is ₦(5,324 - 2,400) = ₦2,924, the total cost is (24 − 0.04x)x = 24x − 0.04x².

<p>True</p> Signup and view all the answers

In determining the value of x in a quadratic equation (model), three approaches are usually used. These are factorization, method of completing the square, and the quadratic formula.

<p>True</p> Signup and view all the answers

The quadratic equation 0.04x² - 24x + 2924 = 0 can be solved using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

<p>True</p> Signup and view all the answers

A wide variety of mathematical methods are used in social science courses and studies. These methods are applied in economics, business, and management studies to address issues such as discounting and investment appraisal.

<p>True</p> Signup and view all the answers

These methods include percentage, series, and sequence. Percentage, series, and sequence are mathematical tools used in finance.

<p>True</p> Signup and view all the answers

The break-even point in business occurs when the total revenue and total costs are equal.

<p>True</p> Signup and view all the answers

If y = f(u) and u = g(x), then y = f(g(x)). Therefore, the chain rule tells that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

<p>True</p> Signup and view all the answers

A limit L is the one and only one finite real number to which the function values ƒ(x) of a function ƒ draw closer for all values of x as x approaches to a, but does not equal to a.

<p>True</p> Signup and view all the answers

A continuous function is one which has no breaks in its curve.

<p>True</p> Signup and view all the answers

A function is differentiable at a point if the derivative exists (may be taken) at that point. To be differentiable at a point, a function must: 1.) Be continuous at that point and 2.) Have a unique tangent line at that point.

<p>True</p> Signup and view all the answers

The slope of a straight line is defined as $S = \frac{\Delta y}{\Delta x}$. In a curve, the slope is taken at a point, and the tangent line at that point is known as the derivative.

<p>True</p> Signup and view all the answers

The derivative of a function f(x) at x is defined as $ƒ’(x) = lim_{\Delta x \to 0} \frac{f(x + \Delta x) − f(x)}{\Delta x}$ if the limit exists.

<p>True</p> Signup and view all the answers

The process of finding the derivative of a function is the differentiation.

<p>True</p> Signup and view all the answers

If f(x) = k, where k is a constant, then f’(x) = 0.

<p>True</p> Signup and view all the answers

If f(x) = mx + b, then f’(x) = m.

<p>True</p> Signup and view all the answers

If f(x) = kxn, then f’(x) = k·n·xn-1.

<p>True</p> Signup and view all the answers

The second-order derivative test, which is applied when the necessary first-order condition is met, is known as the second-order condition.

<p>True</p> Signup and view all the answers

If a function is strictly concave (convex), there will be only one maximum (minimum) and it is called a global maximum (minimum).

<p>True</p> Signup and view all the answers

To be at a relative maximum or minimum at a point a, the function must be at a relative plateau, meaning it is neither increasing nor decreasing at a.

<p>True</p> Signup and view all the answers

A point in the domain of a function where the derivative equals zero or is undefined is known as a critical point or value.

<p>True</p> Signup and view all the answers

If f'(a) = 0 and f''(a) > 0, then the function is convex at a, and hence at a relative minimum.

<p>True</p> Signup and view all the answers

If f'(a) = 0 and f''(a) < 0, then the function is concave at a, and hence at a relative maximum.

<p>True</p> Signup and view all the answers

If f’(a) =0 and f’’(a)=0, then the test is inconclusive.

<p>True</p> Signup and view all the answers

Study Notes

Applied Mathematics for Economics and Social Sciences

  • Course title: Applied Mathematics for Economics and Social Sciences
  • Course code: AEM 303
  • Instructors: R.A. Sanusi and T.O. Oyekale

Classwork Questions

  • Question i: Linear algebra is only applicable to systems of linear equations. (True or False)
  • Question ii: Exponential and power functions can be converted to linear functions and solved via linear algebra. (True or False)
  • Question iii: Rows in a matrix are the numbers in a vertical line. (True or False)
  • Question iv: Columns in a matrix are the numbers in a horizontal line. (True or False)
  • Question v: A square matrix has an equal number of rows and columns. (True or False)
  • Question vi: A column vector is a matrix with a single row. (True or False)
  • Question vii: The dimensions of a column vector are expressed as r x 1. (True or False)
  • Question viii: A row vector has a single column. (True or False)
  • Question ix: The dimensions of a row vector are expressed as 1 x c. (True or False)
  • Question x: The product of two conformable matrices is undefined if the matrices are not conformable for multiplication. (True or False)
  • Question xi: A null matrix is made up entirely of zeros, and can have any number of dimensions. (True or False)
  • Question xii: The coefficient matrix (A) in a system of linear equations contains the variables. (True or False)
  • Question xiii: The solution vector (X) in a system of linear equations contains the coefficients. (True or False)
  • Question xiv: Determinants are single numbers or scalars, and are calculated only for square matrices. (True or False)
  • Question xv: A vanishing determinant is one where |A| = 0. (True or False)
  • Question xvi: A non-singular matrix has a determinant that is not zero. (True or False)
  • Question xvii: If p(A) < n, matrix A is singular and there exists linear dependence. (True or False)
  • Question xviii: A minor matrix is a determinant with a pre-assigned sign. (True or False)
  • Question xix: A triangular matrix has zeros above or below the principal diagonal. (True or False)
  • Question xx: The adjoint matrix is the transpose of a minor matrix. (True or False)
  • Question xxi: The determinant of a matrix equals the determinant of its transpose. (True or False)
  • Question xxii: If all the elements in a row or column of a matrix are zero, the determinant is 1. (True or False)
  • Question xxiii: If two rows or columns in a matrix are linearly dependent, the determinant is zero. (True or False)
  • Question xxiv: A symmetric matrix remains unchanged when transposed. (True or False)
  • Question xxv: An identity matrix in matrix algebra is analogous to the number 1 in ordinary algebra. (True or False)

Review of Mathematical Terms and Concepts

  • Exponents:
    • If n is a positive integer, xn implies x multiplied by itself n times.
    • x = base; n = exponent.
    • Any non-zero number or variable to the power of zero equals 1 (e.g., 30 = 1).
    • Zero to the power of zero is undefined.
    • Rules of exponents: xa(xb) = xa+b; xa/xb = xa-b; (xa)b = xab; (xy)n = xnyn; (x/y)n = xn/yn

Equations and Functions

  • Equation: A mathematical statement stating two expressions equal to each other.
  • Function (f): A rule, assigning each value of a variable (argument of the function) to one and only one value [f(x)].
    • Domain: Set of possible values of x.
    • Range: Set of possible values for f(x).

Graphs

  • To plot a graph y = f(x):
    • x is plotted on the horizontal axis (independent variable)
    • y is plotted on the vertical axis (dependent variable)
    • The graph of a linear function is a straight line.
  • The slope of a line can be expressed as: S = (y – y0) / (x - x0) where ∆y = change in y and ∆x = change in x

Additional Topic Summaries

  • Linear Equations:
    • General form is y=mx+c, where m is the slope and c is the y-intercept.
  • Non-Linear Equations:
    • Do not display a linear relationship.
    • Example is the quadratic equation ay^2+bx+c=0.

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Description

Test your understanding of key concepts in Applied Mathematics for Economics and Social Sciences. This quiz covers fundamental topics such as linear algebra, matrix theory, and vector dimensions. Answer true or false to the statements and gauge your grasp on the subject matter.

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