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Quantitative Methods Session 16 – 22.08.13 Matrices Pranjoy Arup Das

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Matrices or Matrix A matrix is table of numbers listed in a square or rectangular form. Matrices are used in the following applications: Solving linear equations with 2 variables Solving linear equations with 3 variables Set Theory – To solve problems of Relation of sets Cryptology – the study of coding and de-coding secret messages. Input – output analysis Forecasting related studies Transition probability analysis

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This is a 2 x 2 matrix. It has 2 rows and two columns This is a 2 x 3 matrix. It has 2 rows and 3 columns This is a 3 x 4 matrix. It has 3 rows and 4 columns. Matrices are usually denoted by letters such a A, B, C, X etc

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Addition of matrices: Two Matrices having the same number of rows and columns can be added. Suppose A = And B = So A + B = + = = Rule to remember : A + B = B + A

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Multiplication of matrices: Multiplication is only possible if the rows of the 1 st matrix and the column of the 2 nd matrix have the same number of elements. Suppose A = And B = So A * B = * = a = 2 * 5 + 1 * 4 b = 2 * 6 + 1 * 8 c = 3 * 5 + 12 * 4 d = 3 * 6 + 12 * 8 Rule to remember : A * B is not the same as B * A

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Determinant of a matrix: Say A = Then determinant of A or I A I = (ad – bc) Eg. Find the determinant of A = I A I = 5 * 8 – 6 * 4 = 40 – 24 = 16 The determinant of = 16

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Inverse of a matrix: Say A = Inverse is denoted by 1/A or A -1 NOTE : Inverse of a matrix A, i.e. A -1 = Inverse is not possible if (ad- bc) = 0 Eg. Find the inverse of A = A -1 = ==

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Solving simultanoeus linear equations with 2 variables: Suppose we have two equations 2x-y = 3 and 5x+y = 4 We can represent these two equations in the form of 3 matrices: 1)A, which is the co-efficient matrix = 2)X, which is the variable matrix = 3) B, which is the constant matrix = The solution is given by AX = B So =

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Now Since AX = B => X = 1/A * B => X = A -1 * B We know that A -1 = = Using the value of A -1 in X = A -1 * B X = => = = > = So x = 1 and y = -1

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Practice Exercise: 1) Try solving the following three equations by matrix method: 2x + y = 15, 2y + z = 25 & 2z+x = 26 2) Find the inverse of the matrix A = [7 5] [6 6] 3) Find the determinant of the matrix:[7 5] [6 6] 4) Find the product of the two matrices [1 3] & [1 2] [2 2] [2 1] 5) Find the sum of the two matrices: [1 2] & [1 3] [2 1] [2 2]

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Arithmetic Operations on Matrices. 1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row.

Arithmetic Operations on Matrices. 1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row.

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