First Year Commerce: Matrices and Determinants

• Fundamentals of Mathematics by Mayur Kshirsagar, Head, Department of Mathematics and Statistics, Brihan Maharashtra College of Commerce, Pune-04, dated November 9, 2024, includes information for first-year commerce students in the following topics:
• Matrices and Determinant
• Functions
• Limit and Continuity
• Differentiation and its applications

Matrices and Determinant

• Matrix theory is a tool dealing with data as a whole, useful in the theory of equations for solving simultaneous linear equations.
• Determinants relate to matrices and determine unique solutions, applying Cramer's rule for expressing unknowns via equation coefficients.
• Matrix algebra simplifies economics, business, and finance theoretical models while storing a variety of information.
• Matrices help track stock in multiple outlets, reading rows for outlet stock levels and columns for product line stock, represented in matrix form.
• A matrix is a rectangular array of numbers or elements in carefully ordered places.
• Horizontal lines are rows.
• Vertical lines are columns.
• Dimensions are defined as (r × c).
• r equals the number of rows.
• c equals the number of columns.
• The row number always precedes the column number.
• A square matrix has an equal number of rows and columns (r = c).
• A column vector is a matrix with a single column and dimensions r × 1.
• A row vector is a single-row matrix with order 1 × c.
• Matrix equality occurs when two matrices A and B of the same order have the same corresponding elements.
• Matrix addition A + B combines two matrices of the same order by adding corresponding elements
• Matrix subtraction A – B involves subtracting corresponding elements of two matrices.
• Scalar multiplication kA multiplies each matrix element by a scalar k.
• Matrix multiplication of A and B is possible only if the columns in A equal the rows in B
• Matrix A is of m × n order, and matrix B is of n × p order.
• The AB product is defined with the product matrix AB having an m × p order.
• Multiplication of a row vector A by a column vector B requires an equal number of elements in each vector
• The product is found by multiplying individual elements and summing the products
• Matrix addition properties include the commutative law (A + B = B + A), associative law (A + (B + C) = (A + B) + C).
• Existence of identity (A + 0 = 0 + A = A), and existence of inverse A + (-A) = (-A) + A = 0
• Matrix multiplication isn't always commutative (AB = BA is not always true).
• It adheres to the associative law: A(BC) = (AB)C, and the distributive law: A(B + C) = AB + AC and (A + B)C = AC + BC.
• The determinant |A| of a 2×2 matrix, is calculated as a11a22 - a12a21
• This is derived by taking the product of the principal diagonal elements and subtracting the product of off-diagonal elements.
• A square matrix A is non-singular if its determinant is non-zero (|A| ≠ 0). Otherwise, it is a singular matrix.
• A minor of a square matrix element is the determinant with omitted row/column elements.
• A cofactor involves the element's minor with a sign based on its position via the rule (−1)^(i+j)
• Element by the number of row and column.
• An adjoint matrix is the transpose of a cofactor matrix, with elements replaced by their cofactors.
• An inverse matrix A^(-1) that reduces a matrix to an identity matrix A*A^(-1) = I = A^(-1)*A
• The inverse only occurs for square, non-singular matrix A.
  • It is derived by where A^(-1) = 1/|A| * Adj(A)
• A system of linear equations can be solved using an inverse matrix to define the Matrix of unknowns: A*X=b
  • Using inverse method of equation A X=b X = A-1b
• Cramer's rule provides a method for solving AX = b by expressing solutions as ratios of determinants = |Ai|/|A|
• Ai is obtained by replacing the ith column of A with constants.

Functions

• In mathematics, a constant is a quantity that does not change. In comparison, a variable is a quantity that is subject to change
• Variables are commonly denoted by the symbols x, y, z, u, t.
• Variables are related when one variable's value depends on the other.
• A function assigns exactly one value f(x) to each input value (x).
• The domain is the set of all possible input values and the range
• Is the set of all possible values for f(x).
• Functions are often described by algebraic formulas or Greek letters
• Types of Functions
• Linear function: f(x) = mx + b.
• Quadratic function: f(x) = ax² + bx + c, a ≠ 0.
• Polynomial function of degree n: f(x) = anx^n−1 + an−1x^n−1 +...+ a1*x + a0, an ≠ 0.
• Rational function: f(x) = g(x)/h(x).
• Both g(x) and h(x) are polynomial functions and h(x) ≠ 0.
• If denoting y = f(x), then y depends on x. Hence, x is independent variable with y as dependent variable. Composition of functions is defines as f(g(x))
• Addition is expressed as (f + g)(x) = f(x) + g(x)
• Subtraction is expressed as (f - g)(x) = f(x) - g(x)
• Multiplication is expressed as (f × g)(x) = f(x) × g(x)
• Division is expressed as (f/g)(x) = f(x)/ g(x)
• Supply and demand are relationships between the price per unit (p) and the quantity on the market (q).
• The standard technique to find these equations is to solve them simultaneously

Limit and Continuity

• Limit and continuity are real value functions used in calculus
• A Limit looks at defining a functions behaviour near a function, f(x) as -lim x→a(x2−1)/(x−1)
• The function, f(x), is not defined as it approaches 1 through x
• As x approaches 1, f(x) approaches 2
• A function, f(x), approaches L when x = a
• Symbolically, we write the limit of f(x) as x approaches a is L: lim x→a f(x) = L.
• There are two types of limit Two-sided, and One-sided
• One-sided looks at behaviour approaching function form one side
• From the right x→a
  • Or from the left x→a
  • This is because all values can be near a but be greater, or smaller than.
• Two Sided is a function at x, as x approaches a from both the left and the right side
• lim x→a f(x)
• To essentially two ways exist as x approaches a number either from left or from right
• That there are essentially two ways could approach a number a either from left or from right
• If the limit value is called the right-hand limit (RHL)
  • Symbolically, we can write
  • lim x→a+ f(x)
• If it's the left then referred to the left-hand limit (LHL)
  • Symbolically we can write
  • lim x→a- f(x)
• For the limit to exist the LHL and RHL but meet of x at each, otherwise it fails
• We can express in symbolic terms as
  • LHL = lim x→a- f(x)
  • RHS = lim x→a+ f(x)
• The rules of limits, assuming that lim x→a=f(x) and lim x→a=g(x) both exits, the rules of limits are given below
• lim x→a : k = k, where k is a constant
• lim x→a : xn = an
• lim x→a : k f(x) = k lim x→a f(x), where k is a constant
• lim x→a : (f(x) + g(x)) = lim x→a f(x) + lim x→a g(x)
• lim x→a : (f(x) - g(x)) = lim x→a f(x) – lim x→a g(x)
• lim x→a : (f(x) × g(x)) = lim x→a f(x) × lim x→a g(x)
• lim x→a : (f(x)/ g(x)) =lim x→a f(x)/lim x→a g(x), where lim g(x) ≠ 0
• lim x→a : (f(x))n = [(lim x→a f(x)]^n
• Continuity occurs when the graph function is continous without any breaks or jumps such that;
• A continuous function is one which has no breaks in its curve. It can be drawn without lifting the pencil from the paper
• That f(x) is defined, that is, value of function exists, at x = a
• That lim x→1 f(x) exists
• That lim x→1 f(x) = f(a) Properties of continous functions
• If f and g are are continous functions then:
• f+g, f-g, f*g = continuous at x=a
• f/g is a continuous function proved g(a)!=0 - ALL Polynomial functions are continous as ALL real numbers
  • If f,g are continous functions then f o g is continous

Differentiation and its applications

• Derivatives, also known as slope, look at the change of of a function (y) to an independent variable (x)
• Dy/Dx = lim dy/dx
• A tangent line is tangent to the the slope from said point Geometrically, slope of a curvilinear function at given point is where tangent is
• Letting these changes occur, we can identify it's slope, using limits as it approaches the tangent;
• Slope equation = f(x+∆x ) == f(x) / ∆x
• From this the key formula can be used to determine the derive
  • f'(x) = lim h→0 = f(x + h) - (x) /h Rules of differentiation
• Constant function, derivative = zero
• f(x) = k then f'(x) = 0
• Function for typef(x) = ax + b, or linear function = given by f'(x) = a
  • if f(x) = ax +b then f'(x) = a
• The generalized power rule of a function that determines that;
• If f(x) = ((X(x)]^n, where glx) to is differential function and n is any real number then;
• 1x) = n [g(x)1-1 g‘(x)
• Power Function rule of dervatives f(x)=Kx^n; k =constant , n = any real number is given as;
• f'(x) =knx^n-1
• The sum or product rule allows dervatives to use sum of differentable functions;
• if f(x)=g(x)+ h(x) then:
• f’(x) = g‘(x) + h‘(x)
• f’(x) = g‘(x) - h(x)
• The Product rule: where g(x) and h(x) are both differentable functiions, equal to first function Multiplied by dervative of second and more
• This equates to;
  • h(x).g'(x)
• glx) + h(x)
• The quotient involves differentable functions and A(x) 1s given by following formula;
• Equating:
  • g'(x)h(x) – h(x)g(x) / (h(x)]^2 The increasing/decreasing function determine the slope/vicinity of a points derivative;
• The point (a, f(a)) looks at what is rising and falling over the over gradient
• Derivative is positive at x, equates to increase to a function

Maximum and minimum of function

• A relative extremum is a point at which a function is at a relative maximum or