# Functions, Differentiation,Matrix

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### Functions :

Functions 8/5/2009 1 Manavazhaganr@bsnl.in

### Constants , Variables & Functions :

Constants , Variables & Functions What is the value of 5 ? X ? Profit ? Sales ? 7+8 ? Imports ? 8/5/2009 2 Manavazhaganr@bsnl.in

### Costants , Variables & Functions :

Costants , Variables & Functions Constant A quantity which does not change its value through out a set of mathematical operations. Constants are denoted as a, b, c etc. Variable A quantity which can assume any value out of any set of given values. Variables are denoted as x, y, z etc. Exercise 8/5/2009 3 Manavazhaganr@bsnl.in

### How to express the Relationship ? :

How to express the Relationship ? Relationship between Hours of study and Marks obtained Relationship between Number of advertisements and Sales Relationship between Expenditure and Profit Relationship between Rain and Crop yield FUNCTIONS Exercise 8/5/2009 4 Manavazhaganr@bsnl.in

### Functions :

Functions Function describes how the value of one quantity depends on the value of another quantity. Function is a rule which assigns to each number from one set to exactly only one number in another set. Mark obtained depends on the hrs. of study Mark – Dependent variable Hrs. of Study – Independent Variable Mark is the function of Hrs. of study Mark = f ( Hrs. of study ) 8/5/2009 5 Manavazhaganr@bsnl.in

### Formulating a function :

Formulating a function Mark = f ( 10 times of Hrs. of study ) Assume Mark= y Hrs. of study = x then formula for the Mark and Hrs. of study relation is Mark = f ( 10 times of Hrs. of study ) y = f (x) = 10x or f(x) = 10x 8/5/2009 6 Manavazhaganr@bsnl.in

### Dependent & Independent Variables - Quiz :

Dependent & Independent Variables - Quiz Find the dependent , Independent variable & Formula for function : Number of advertisements and Sales Expenditure and Profit Rain and Crop yield Exercise 8/5/2009 7 Manavazhaganr@bsnl.in

### Domain & Range of Functions :

Domain & Range of Functions Function is a rule which assigns to each number from one set to exactly only one number in another set. Y= 10 X 8/5/2009 8 Manavazhaganr@bsnl.in

### Domain & Range of Functions - Quiz :

Domain & Range of Functions - Quiz If Domain= 0,1,2,3,4 Find the range of Y= 3x+1 If Domain= 0,1,2,3,4 Find the range of Y= 2x-5 Exercise 8/5/2009 9 Manavazhaganr@bsnl.in

### Functions -Quiz :

Functions -Quiz Examples of Functions Y= 2x +3 Y= 0.3x-7 F(x)= ax+b F(x)= -89x + 90 Whether the following are functions or not Y= -6x+1 Y < x + 2 Y= X2 X = Y2 - 1 Y > x + 2 Exercise 8/5/2009 10 Manavazhaganr@bsnl.in

### Functions - Exercise :

Functions - Exercise 3. A contractor estimates that the total cost of building x large houses is approximated by A(x) = X2 + 80x + 60 A(x) is the cost in thousand rupees Find the cost of building 4 Houses 10 houses 1. Let f (x)= -3x-8 find 1. f (0) 2. f(-1) 3. f(5) 2. Let f (x)= 5X2 -2x +1 find 1. f (1) 2. f(3) 3. f(m) Exercise 8/5/2009 11 Manavazhaganr@bsnl.in

### Graphical representation of Function :

Graphical representation of Function Hr. of study X Y Marks Obtained Exercise Y= 10 X 8/5/2009 12 Manavazhaganr@bsnl.in

### Which of the following is Function ? - Quiz :

Which of the following is Function ? - Quiz 8/5/2009 13 Manavazhaganr@bsnl.in

### Which of the following is Function ? :

Which of the following is Function ? If a vertical line cuts the graph in more than one point, then the graph is not the graph of a function 8/5/2009 14 Manavazhaganr@bsnl.in

### Types of Functions :

Types of Functions Linear Functions Quadratic Functions Polynomial Functions Exponential Functions Logarithmic Functions 8/5/2009 15 Manavazhaganr@bsnl.in

### Structure of Different Types of Functions :

Structure of Different Types of Functions Exercise 8/5/2009 16 Manavazhaganr@bsnl.in

### Linear Functions - Quiz :

Linear Functions - Quiz Any equation of the form Y= mX+c is a linear function. This function forms a straight line Find out whether the following are linear equations or not Y= 5x-8 Y= -6-4x Y= -x Y= -x-5 + 2g -7b Y= 8 Y= xy+c X=y-5c Exercise 8/5/2009 17 Manavazhaganr@bsnl.in

### Sketching Linear Functions :

Sketching Linear Functions Two points are required to s Two points are required to draw a Linear function 8/5/2009 18 Manavazhaganr@bsnl.in

### Sketching Linear Function :

Sketching Linear Function Exercise 8/5/2009 19 Manavazhaganr@bsnl.in

### Sketching Linear Functions :

Sketching Linear Functions Two points are required to s These two points are called X- intercept , y-intercept . (2,20),(4,40), (6,60) The numbers that multiply x & y are called Coefficients 8/5/2009 20 Manavazhaganr@bsnl.in

### Find Intercepts & Linear Equation :

Find Intercepts & Linear Equation Y = 2X + 5 Y= 2x 8/5/2009 21 Manavazhaganr@bsnl.in

### Interpretation of Linear Functions :

Interpretation of Linear Functions Any equation of the form Y= aX+b is a linear function. This function forms a straight line 8/5/2009 22 Manavazhaganr@bsnl.in

### Interpretation of Linear Functions :

Interpretation of Linear Functions Y= aX + b Y2 – y1 8/5/2009 23 Manavazhaganr@bsnl.in

### Find the gradient of the following Functions - Quiz :

Find the gradient of the following Functions - Quiz Y = 2x + 3 Y = -4x – 2 Y = 2x Y =2 2x + 3y = 6 Exercise 8/5/2009 24 Manavazhaganr@bsnl.in

### Applications of Linear Functions :

Applications of Linear Functions Cost Function Revenue Function Profit Function Break-Even Point 8/5/2009 25 Manavazhaganr@bsnl.in

### Applications of Linear Functions :

Applications of Linear Functions Cost Function Cost function is the cost of producing x units of goods Cost function C(x) = Fixed cost + variable cost Suppose that the cost of producing a clock can be approximated by the linear function C (x) = 12 x + 100, C(x) is the cost in rupees to produce x clocks. Find the cost of producing : 1. 0 clock 2. 1 clock 3. 2 clocks 4. 3 clocks 5. 4th clock 5. Marginal cost Cost of production = Variable cost + Fixed Cost Cost of 0 clock: C(0) = 12*0 +100 = Rs. 0 + 100 Cost of 1 clock: C(1) = 12*1 +100 = Rs.12 + 100 Cost of 2 clocks: C(2) = 12*2 +100 = Rs.24 + 100 Cost of 3 clocks: C(3) = 12*3 +100 = Rs.36 + 100 Marginal cost = cost of producing one additional clock = Variable cost = Slope of the linear equation = 12 Marginal cost = Variable cost = Slope =12 Fixed cost = Y- intercept = 100 Exercise 8/5/2009 26 Manavazhaganr@bsnl.in

### Applications of Linear Functions :

Applications of Linear Functions Cost Function - Exercise Write the cost function , variables: Car rental firm charges Rs. 12 plus Rs.1 per hour Taxi charges Rs. 45 plus Rs. 2 per mile Parking charges Rs. 3 plus Rs. 5 per hour Cost of production = Variable cost + Fixed Cost Y= 12 +x Variable cost =1 Marginal cost =1 Fixed cost = 12 Y= 2x+45 Variable cost = 2 Marginal cost =2 Fixed cost = 45 Y= 3 + 5x Variable cost = 5 Marginal cost = 5 Fixed cost = 3 Exercise 8/5/2009 27 Manavazhaganr@bsnl.in

### Applications of Linear Functions :

Applications of Linear Functions Revenue Function Total amount of money generated from the sale of x units of goods. Revenue R (x) = Sales * Price Profit Function Total amount of money available from sales after deducting cost Profit = Revenue- cost Break-Even point Company make no profit or loss Revenue = Cost For a certain magazine Cost of production is C(x)= 0.20 x +1200, Revenue is R(x)= 0.5x where x is the number of magazine sold. Find the Break-Even point C(x) = 0.20 x + 1200 R(x) = 0.50 x Break-even point will be at C(x) = R(x) C(x) = R(x) 0.20x+1200 = 0.50x 1200= 0.30x x = 1200/ 0.3 x= 4000 The company should sell 4000 magazines to break-even Exercise 8/5/2009 28 Manavazhaganr@bsnl.in

Quadratic Functions Any equation of the form Y= aX2+bX+c is a Quadratic function. This function forms a Parabola Find out whether the following are quadratic equations or not Y= 5x-8 Y= -6-aX2 Y= X2 Y= -x-5X2 Y= 8X3-5X2-x +8 Y= 6890 -5X2-x = y-5c Exercise 8/5/2009 29 Manavazhaganr@bsnl.in

Sketching Quadratic Functions Y = X2 Y = -(X+2)2 Exercise 8/5/2009 30 Manavazhaganr@bsnl.in

### How do Quadratic Equations arise ? :

How do Quadratic Equations arise ? Revenue R (x) = Sales * Price If price = x-2 then R (x) = sales (x ) * (x-2) R ( x) = -2x + X2 8/5/2009 31 Manavazhaganr@bsnl.in

Quadratic Equations The curve crosses the X axis at Solve -3x +2 + X2 A= 1, b= -3 , c=2 The Quadratic curve turns at : Exercise X= 2 8/5/2009 32 Manavazhaganr@bsnl.in

### Differentiation :

Differentiation 8/5/2009 33 Manavazhaganr@bsnl.in

### Applications of Differentiation :

Applications of Differentiation Marginal Cost Marginal Revenue Maximum Profit Minimum cost 8/5/2009 34 Manavazhaganr@bsnl.in

### Rates of Change :

Rates of Change Function Relationship between Hours of study and Marks obtained Relationship between Number of advertisements and Sales Relationship between Expenditure and Profit Relationship between Rain and Crop yield Rates of Change How Marks obtained changes with respect to Hours of study How sales changes with respect to Number of advertisements How profit changes with respect to Expenditure How Crop yield changes with respect to Rain 8/5/2009 35 Manavazhaganr@bsnl.in

### Differentiation & Derivative :

Differentiation & Derivative The process of finding derivatives of a function is called differentiation The gradient of a function is Derivative. The rate at which the value of a function changes with reference to the independent variable is Derivative Change in y Change in x Derivative is written as dy dx Y2 – y1 8/5/2009 36 Manavazhaganr@bsnl.in

### Finding Derivative Graphically :

Finding Derivative Graphically Derivative is 2x - 6 - 4 - 2 2 4 6 Slope at x=2 is : (8-0)/(2-0) =4 The gradient of a function is Derivative. The slope of the tangent to the function is Derivative. 8/5/2009 37 Manavazhaganr@bsnl.in

### The Easy way :

The Easy way 8/5/2009 38 Manavazhaganr@bsnl.in

### Rules of Differentiation – First Order :

Rules of Differentiation – First Order Derivative of a constant c = 0 c - is constant 2. Power Rule 3. Product Rule 5. Quotient Rule 6. Sum Rule Exercise ( U, V are not constants ) 4. Multiple Rule ( U is a constant ) ( U, V are not constants ) ( U, V are not constants ) 8/5/2009 39 Manavazhaganr@bsnl.in

### Rule-1: Derivative of a constant- Quiz :

Rule-1: Derivative of a constant- Quiz Rule-1: Derivative of a constant c = 0 c - is constant Find the derivative of the Following: 1. 2590 2. 33.45 3. - 790 4. 0.09 5. -800 (2590 ) = 0 8/5/2009 40 Manavazhaganr@bsnl.in

### Rule-2: Power Rule : Quiz :

Rule-2: Power Rule : Quiz Rule-2: Power Rule Find the derivative of the Following: X2 X3 X4 X5 X X-3 2X3 1/x2 -2X-4 904 n = (+ ) 2 n = ( + ) 3 n = (- ) 3 8/5/2009 41 Manavazhaganr@bsnl.in

### Rule-3: Sum Rule : Quiz :

Rule-3: Sum Rule : Quiz Find the derivative of the Following: X2 + X3 X3 - X4 X4 + 1/x2 X5 - X-3 904 - 2X-4 { U = (+ ) X2 , V = (+ ) X3 } Step-1 : Find out the Format Step-2 : Apply the suitable formula Step-3 : Apply the power rule on thr right side = 2x + = 2x + 3 X2 8/5/2009 42 Manavazhaganr@bsnl.in

### Rule-3: Sum Rule : Quiz :

Rule-3: Sum Rule : Quiz Find the derivative of the Following: X2 + X3 X3 - X4 X4 + 1/x2 X5 - X-3 904 - 2X-4 { U = (+ ) X3 , V = (- ) X4 } Step-1 : Find out the Format Step-2 : Apply the suitable formula Step-3 : Apply the power rule on thr right side = 3 X2 - = 3 X2 _ 4 X3 8/5/2009 43 Manavazhaganr@bsnl.in

### Differentiation - Applications :

Differentiation - Applications Marginal Revenue Marginal Cost Rate of change Maximum Profit Minimum Cost Slope of the tangent line 8/5/2009 44 Manavazhaganr@bsnl.in

### Differentiation - Applications :

Differentiation - Applications Marginal Cost The cost of manufacturing x thousand barrels of beer in rupee is C(x) = 4 X2 + 100x + 500 , find the marginal cost of manufacturing 5 thousand barrels of beer. Step-1: Find the derivative of cost function C(x) = 8X + 100 Step- 2: Substitute x with 5 C’(5) = 8*5 + 100 = 140 The cost of producing 5 000 barrels of beer is Rs. 140/- 8/5/2009 45 Manavazhaganr@bsnl.in

### Differentiation - Applications :

Differentiation - Applications Marginal Cost The cost of manufacturing x Radiators in rupee is C(x) = X3 -6X2 + 15x when 8 to 30 radiators are produced. Now your shop produces 10 radiators a day. How much extra will it cost to produce one more radiator a day ? Step-1: Find the derivative of cost function C(x) = 3X2 -12X + 15 Step- 2: Substitute x with 10 C’(5) = 3*100 – 12(10) + 15 = 195 The additional cost will be Rs. 195/- 8/5/2009 46 Manavazhaganr@bsnl.in

### Differentiation - Applications :

Differentiation - Applications Marginal Revenue The revenue from the sale of x thousand barrels of beer in rupee is R(x) = 10 X2 + 800x , find the marginal revenue of selling 5 thousand barrels of beer. Step-1: Find the derivative of cost function C’(x) = 20x + 800 Step- 2: Substitute x with 5 C’(5) = 20*5 + 800 = 900 The revenue in selling 5 000 barrels of beer is Rs. 900/- 8/5/2009 47 Manavazhaganr@bsnl.in

### Finding Maximum & Minimum :

Finding Maximum & Minimum A function f (x) is maximum at x=c if f’(c) =0 and f’’ ( c ) < 0 A function f (x) is minimum at x=c if f’(c) =0 and f’’ ( c ) > 0 Step-1 : Find the derivative of the function : ( f’(x) ) Step-2 : Find the x value by equating f’(x) = 0 Step- 3 : Again Find the derivative of f’(x) : ( f’’(x) ) Step- 4 : If the second derivative ( f’’(x) ) is negative , then the function f(x) will be maximum at the value of x found at step-2. Or else if f’’(x) ) is positive , then the function (x) will be minimum at the value of x found at step-2. Step- 5 : Maximum value or Minimum value of the function can be found by substituting x value found in step -2 in the original function f ( x ) 8/5/2009 48 Manavazhaganr@bsnl.in

### Finding Maximum Profit- Exercise :

Finding Maximum Profit- Exercise Step-1: P’(x) = 80 -2x Step-2 : 0 = 80- 2x = 4 X = 40 Step-3 : P’’(x) = -2 Step-4 : P’’(x) is negative . Hence Profit will be maximum at 40 advertisements Step-5 : The is : P(40) = 120+80 *40 – (40*40) Maximum Profit = Rs. 1720 The profit P(x) due to advertising x numbers , in rupees is given by P(x)= 120+80x- X2. What amount of advertising fetches maximum profit ? What is the maximum profit ? 8/5/2009 49 Manavazhaganr@bsnl.in

### Finding Minimum Cost - Exercise :

Finding Minimum Cost - Exercise Step-1: C’(x) = 2X-1200 Step-2 : 0 = 2x -1200 X = 600 Step-3 : C’’(x) = 2 Step-4 : C’’(x) is Positive . Hence cost will be minimum if we produce 600 balls Step-5 : The cost per ball if we produce 600 balls is : C(600) = (600*600)-1200 (600) +360040 Cost per ball if we produce 600 balls is : = Rs. 40 per ball The cost of manufacturing cricket ball is C(x)= X2 - 1200X+ 360040. x is number of balls produced. How many balls should be produced at which the cost is minimum ? What is the cost per ball at this production level ? 8/5/2009 50 Manavazhaganr@bsnl.in

### Integration :

Integration 8/5/2009 51 Manavazhaganr@bsnl.in

### Integration :

Integration Integration is the reverse process of differentiation If f(x) is a function having derivative f’(x) then antiderivative of f’(x) is f(x) Derivative of cost function is marginal cost function, so the antiderivative of marginal cost function is cost function The antiderivative of a function f is written as : ∫ f(x) dx The process of finding ∫ f(x) dx Is called Integration. ∫ - Integral sign F(x) - integrand dx – indicates we are integrating with respect to x 8/5/2009 52 Manavazhaganr@bsnl.in

### Rules of Integration :

Rules of Integration ∫xn dx = xn+1 + C n+1 ∫ k * f(x) dx = K* ∫f(x)dx ∫ f(x) + g(x) dx = ∫f(x)dx + ∫g(x)dx ∫x-1 dx = ∫ 1/ x dx 8/5/2009 53 Manavazhaganr@bsnl.in

### Integration - Exercise :

Integration - Exercise ∫xn dx = xn+1 + C n+1 Find ∫x3 dx ∫x3 dx = x3+1 + C 3+1 ∫x3 dx = x4 + C 4 Find ∫ dx ∫x0 dx = x0+1 + C 0+1 ∫x0 dx = x1 + C 1 ∫x0 dx = x + C 8/5/2009 54 Manavazhaganr@bsnl.in

### Applications of Integration :

Applications of Integration Finding Cost function using Marginal Cost function Finding Revenue function using Marginal Revenue function Maximum Profit when marginal revenue is given 8/5/2009 55 Manavazhaganr@bsnl.in

### Matrices :

Matrices 8/5/2009 56 Manavazhaganr@bsnl.in

### Matrices :

Matrices A Matrix (Matrices- Plural ) is a rectangular arrangement of numbers enclosed in brackets. Each number inside the matrix is called an element A matrix is referred to by a capital letter A= Example A company manufactures sofas and chairs in three models A,B,C. The company manufactures 10 model A sofas, 12 model B sofas , 5 model C sofas, 15 model A chairs, 20 model B chairs, 8 model C chairs in August. How to organise ? Sofas 10 mod-A 12 mod-B 5 mod C Chairs 15 mod-A 20 mod-B 8 mod C M= Matrix M contains 2 rows & 3 columns M’s dimension is “ 2 by 3 “ Order of M is 2 X 3 8/5/2009 57 Manavazhaganr@bsnl.in

### Matrices - Quiz :

Matrices - Quiz What is the order of the following matrices? 1. 2 by 2 2. 3 by 3 3. 1 by 3 4. 4 by 1 Row matrix Matrix having one row Column matrix Matrix having one Column Equal matrices Matrices should have same order And Corresponding pairs of elements should be equal = 8/5/2009 58 Manavazhaganr@bsnl.in

Addition of Matrices Rule Only matrices with same order can be added or subtracted Example A = B= Find the sum of A &B A+B = 8/5/2009 59 Manavazhaganr@bsnl.in

### Subtraction of Matrices :

Subtraction of Matrices Rule Only matrices with same order can be added or subtracted Example A = B= Subtract A from B A-B = 8/5/2009 60 Manavazhaganr@bsnl.in

### Addition & Subtraction of Matrices- Quiz :

Addition & Subtraction of Matrices- Quiz Example A = B= Find the sum of A &B Example A = B= Subtract A from B Only matrices with same order can be added or subtracted 8/5/2009 61 Manavazhaganr@bsnl.in

### Multiplication of Matrices :

Multiplication of Matrices Example A = B= Find the product of A &B Step-1 : Verify no. of columns of A & rows of B Columns of A = 3 Rows of B= 3 Step-2 : Multiply elements of first row of A and the corresponding elements of the first column of B and add Rule : Number of Columns of A should be equal to number of rows of B + + Step-3 : Multiply elements of first row of A and the corresponding elements of the second column of B and add Step-4 : Multiply elements of Second row of A and the corresponding elements of the first column of B and add The result matrix will have as many rows as A and as many columns as B (2*8 ) + (3*2) + (5*5) = 47 A x B = (2*6 ) + (3*8) + (5*-2) = 26 = = 8/5/2009 62 Manavazhaganr@bsnl.in

### Multiplication of Matrices :

Multiplication of Matrices Step-5 : Multiply elements of second row of A and the corresponding elements of the second column of B and add Step-6 : Multiply elements of third row of A and the corresponding elements of the first column of B and add Rule : Number of Columns of A should be equal to number of rows of B Step-7 : Multiply elements of third row of A and the corresponding elements of the second column of B and add The result matrix will have as many rows as A and as many columns as B = = = Example-2 2 x = 8/5/2009 63 Manavazhaganr@bsnl.in

### Matrix Multiplication- Quiz :

Matrix Multiplication- Quiz Which of the following matrices can be multiplied A is 2x2 B is 2x2 A is 3x2 B is 2x3 A is 4x5 B is 5x2 A is 7x3 B is 2x7 A is 4x2 B is 3x4 A is 432x231 B is 231x4 What is the order of the resultant matrix of the following multiplication A is 4x2 B is 2x4 AxB = 4x2 matrix A is 5x5 B is 2x5 AxB = not possible A is 3x7 B is 7x4 AxB = 3x7 matrix A is 9x8 B is 8x4 AxB = 9x8 matrix A is 800x 999 B is 999x4 AxB = 800x999 matrix Rule : Number of Columns of A should be equal to number of rows of B The result matrix will have as many rows as A and as many columns as B 8/5/2009 64 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a Matrix 1.Determinant of a Matrix Determinant of a matrix A is written as |A| Determinant of 2x2 matrix Determinant of 3x3 matrix A = |A | = |A | = a11*a22 – a12*a21 A = |A | = |A | = a11 - a12 + a13 Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix 8/5/2009 65 Manavazhaganr@bsnl.in

### Determinant of a Matrix - Quiz :

Determinant of a Matrix - Quiz 8/5/2009 66 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a Matrix 2.Minor of a Matrix A minor of an element aij is denoted by Mij is a sub-determinant of |A| obtained by eleminating ith row and jth column. 3.Cofactor of a Matrix Cofactor is a minor with sign It is written as Cij = (-1)i+jMij C11 = (-1)1+1M11 = (-1)2M11 = C12 = (-1)1+2M12 = (-1)3M12 = - M11 = = A = M12 = Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix 8/5/2009 67 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a Matrix 2.Minor of a Matrix A minor of an element aij is denoted by Mij is a sub-determinant of |A| obtained by eleminating ith row and jth column. M11 = = A = M12 = Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix Minor of a Matrix – Quiz 8/5/2009 68 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a Matrix Cofactor of a matrix - Quiz Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix 3.Cofactor of a Matrix Cofactor is a minor with sign It is written as Cij = (-1)i+jMij C11 = (-1)1+1M11 = (-1)2M11 = C12 = (-1)1+2M12 = (-1)3M12 = - 8/5/2009 69 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a Matrix 4.Transpose of the Matrix Transpose of a matrix A is denoted by AT AT = A= Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix 8/5/2009 70 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

4.Adjoint of a Matrix A matrix obtained by replacing its elements with their corresponding cofactors is called Adjoint matrix. Noted as Adj.(A) Step-2: Substitute Cofactors in A Adj.(A) = Inverse of a Matrix Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix A= Step-1: Find the Cofactors of A Cofactor of 2 = (-1)1+2 * Minor of 2 Cofactor of 8 = (-1)1+1 * Cofactor of 8 = 4 Cofactor of 2 = -6 Cofactor of 6 = Cofactor of 4 = 8 Cofactor of 8 = (-1)1+1 * Minor of 8 (-1)2+1 * Minor of 6 Cofactor of 6 = -4 Cofactor of 4 = (-1)2+2 * Minor of 4 4 8/5/2009 71 Manavazhaganr@bsnl.in

### Inverse of a Matrix :

Inverse of a matrix A is A-1 = Adj A Adj A = Inverse of a Matrix Determinant of the Matrix Minor of the Matrix Cofactor of the Matrix Adjoint of the Matrix |A| A= Example |A| = 32-12 = 20 Step-2 : Find the Adj.(A) Step-1 : Find the Determinant of A Step-3 : Apply Inverse formula A-1 = 1/20 A-1 = A-1 = 8/5/2009 72 Manavazhaganr@bsnl.in