Applied Mathematics AEM 303 Quiz

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

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

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

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

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

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

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

A row vector is a matrix having a single column.

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

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

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

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

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

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

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

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

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

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

A minor matrix is a determinant with a prescribed sign.

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

An adjoint matrix is the transpose of a minor matrix.

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

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

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

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

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

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).

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

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

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

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.

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.

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.

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

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

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

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.

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

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.

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.

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.

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.

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

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.

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

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

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

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.

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

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.

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.

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.

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

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

If f(x) = mx + b, then f’(x) = m.

If f(x) = kxn, then f’(x) = k·n·xn-1.

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

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

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.

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

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

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

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