Explanation Given: Democratic vote’s percentage in this election is D = 42 % , Republican vote’s percentage is R = 42 % and Consumer party vote’s percentage is C = 16 % . Procedure used: 1. Form an augmented matrix [ A | I ] where a matrix A is placed along with a same dimension identity matrix I separated by a line. 2. Apply row operations on [ A | I ] in order to obtain another augmented matrix [ I | B ] where I , is again an identity matrix and B is an inverse of A. 3. If the reduction process yields a row of zeros in the left half of the augmented matrix, then the inverse of A does not exist. Calculation: Take A = [ 50 40 30 40 50 30 10 10 40 ] and X = [ d r c ] and transform [ 50 40 30 40 50 30 10 10 40 ] [ d r c ] = [ D R C ] as follows. A − 1 A X = A − 1 [ 42 42 16 ] ( Multiply with A − 1 on RHS of both sides ) I X = A − 1 [ 42 42 16 ] ( ∵ A − 1 A = I ) X = A − 1 [ 42 42 16 ] Obtain the augmented matrix as follows. [ 50 40 30 1 0 0 40 50 30 0 1 0 10 10 40 0 0 1 ] Apply the row operations 40 R 1 − 50 R 2 → R 2 and R 1 − 5 R 3 → R 3 and reduce the matrix. [ 50 40 30 1 0 0 0 − 900 − 300 40 − 50 0 0 − 10 − 170 1 0 − 5 ] Apply the row operations 22.5 R 1 + R 2 → R 1 and R 2 − 90 R 3 → R 3 and reduce the matrix. [ 1 , 125 0 375 62.5 − 50 0 0 − 900 − 300 40 − 50 0 0 0 15 , 000 − 50 − 50 450 ] Apply the row operations 40 R 1 − R 3 → R 1 and 50 R 2 + R 3 → R 2 and reduce the matrix

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Chapter 7.4, Problem 28E

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To find: The matrix [drc] that represents the last election votes of three parties such that [504030405030101040][drc]=[DRC] gives the votes percentage in this election.

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Chapter 7.4, Problem 27E

Chapter 7.4, Problem 29E

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The matrix [ d r c ] that represents the last election votes of three parties such that [ 50 40 30 40 50 30 10 10 40 ] [ d r c ] = [ D R C ] gives the votes percentage in this election.

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To find: The matrix [drc] that represents the last election votes of three parties such that [504030405030101040][drc]=[DRC] gives the votes percentage in this election.

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