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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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3.5 Solution by Determinants. The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.

3.5 Solution by Determinants. The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.

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