Explanation Definition: The procedure for finding the inverse of a square matrix A : Step 1: Augment A with I ; that is, write A | I , where I is the identity matrix of the same order as A . Step 2: Perform elementary row operations using Gauss-Jordan elimination to change the matrix A into the identity matrix I , if possible. Step 3: If at any time you obtain all zeros in a row or column to the left of the dividing line, then there will be no inverse. Step 4: If steps 1 and 2 can be performed, the result in the augmented part is the inverse of A . Calculation: Let A = P + I = 2 0 0 0 0 1 2 0 0 0 1 2 2 0 0 1 3 3 2 0 1 4 6 4 2 . Write the augmented matrix B = A I . B = 2 0 0 0 0 1 2 0 0 0 1 2 2 0 0 1 3 3 2 0 1 4 6 4 2 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Divide row 1 by 2. → 1 0 0 0 0 1 2 0 0 0 1 2 2 0 0 1 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Row 2= row 2 - row 1. → 1 0 0 0 0 1 - 1 2 - 0 0 - 0 0 - 0 0 - 0 1 2 2 0 0 1 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 0 - 1 2 1 - 0 0 - 0 0 - 0 0 - 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 = 1 0 0 0 0 0 2 0 0 0 1 2 2 0 0 1 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Row 3= row 3 - row 1. → 1 0 0 0 0 0 2 0 0 0 1 - 1 2 - 0 2 - 0 0 - 0 0 - 0 1 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 0 - 1 2 0 - 0 1 - 0 0 - 0 0 - 0 0 0 0 1 0 0 0 0 0 1 = 1 0 0 0 0 0 2 0 0 0 0 2 2 0 0 1 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 - 1 2 0 1 0 0 0 0 0 1 0 0 0 0 0 1 Row 4= row 4 - row 1. → 1 0 0 0 0 0 2 0 0 0 0 2 2 0 0 1 - 1 3 - 0 3 - 0 2 - 0 0 - 0 1 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 - 1 2 0 1 0 0 0 - 1 2 0 - 0 0 - 0 1 - 0 0 - 0 0 0 0 0 1 = 1 0 0 0 0 0 2 0 0 0 0 2 2 0 0 0 3 3 2 0 1 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 - 1 2 0 1 0 0 - 1 2 0 0 1 0 0 0 0 0 1 Row 5= row 5 - row 1. → 1 0 0 0 0 0 2 0 0 0 0 2 2 0 0 0 3 3 2 0 1 - 1 4 - 0 6 - 0 4 - 0 2 - 0 1 2 0 0 0 0 - 1 2 1 0 0 0 - 1 2 0 1 0 0 - 1 2 0 0 1 0 0 - 1 2 0 - 0 0 - 0 0 - 0 1 - 0 = 1 0 0 0 0 0 2 0 0 0 0 2 2 0 0 0 3 3 2 0 0 4 6 4 2 1 2 0 0 0 0 - 1 2 1 0 0 0 - 1 2 0 1 0 0 - 1 2 0 0 1 0 - 1 2 0 0 0 1 Divide row 2 by 2. → 1 0 0 0 0 0 1 0 0 0 0 2 2 0 0 0 3 3 2 0 0 4 6 4 2 1 2 0 0 0 0 - 1 4 1 2 0 0 0 - 1 2 0 1 0 0 - 1 2 0 0 1 0 - 1 2 0 0 0 1 Row 3=row 3 - 2 ( row 2). → 1 0 0 0 0 0 1 0 0 0 0 - 2 ( 0 ) 2 - 2 ( 1 ) 2 - 2 ( 0 ) 0 - 2 ( 0 ) 0 - 2 ( 0 ) 0 3 3 2 0 0 4 6 4 2 1 2 0 0 0 0 - 1 4 1 2 0 0 0 - 1 2 - 2 - 1 4 0 - 2 1 2 1 - 2 ( 0 ) 0 - 2 ( 0 ) 0 - 2 ( 0 ) - 1 2 0 0 1 0 - 1 2 0 0 0 1 = 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 3 3 2 0 0 4 6 4 2 1 2 0 0 0 0 - 1 4 1 2 0 0 0 0 - 1 1 0 0 - 1 2 0 0 1 0 - 1 2 0 0 0 1 Row 4=row 4 - 3 ( row 2). → 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 - 3 ( 0 ) 3 - 3 ( 1 ) 3 - 3 ( 0 ) 2 - 3 ( 0 ) 0 - 3 ( 0 ) 0 4 6 4 2 1 2 0 0 0 0 - 1 4 1 2 0 0 0 3 2 - 2 1 0 0 - 1 2 - 3 - 1 4 0 - 3 1 2 0 - 3 ( 0 ) 1 - 3 ( 0 ) 0 - 3 ( 0 ) - 1 2 0 0 0 1 = 1 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 3 2 0 0 4 6 4 2 1 2 0 0 0 0 - 1 4 1 2 0 0 0 0 - 1 1 0 0 1 4 - 3 2 0 1 0 - 1 2 0 0 0 1 Row 5=row 5 - 4 ( row 2)

13th Edition

ISBN: 9781285697734

Author: karl J. smith

Publisher: Cengage Learning US

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Chapter 16.4, Problem 56PS

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The inverse of the Pascal’s matrix.

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Chapter 16.4, Problem 55PS

Chapter 16.4, Problem 57PS

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Chapter 16 Solutions

Nature of Mathematics

Ch. 16.1 - IN YOUR OWN WORDS Can you relate to the Fox Trot...Ch. 16.1 - Prob. 2PS Ch. 16.1 - Prob. 3PS Ch. 16.1 - Prob. 4PS Ch. 16.1 - Prob. 5PS Ch. 16.1 - Prob. 6PS Ch. 16.1 - Prob. 7PS Ch. 16.1 - Prob. 8PS Ch. 16.1 - Prob. 9PS Ch. 16.1 - Prob. 10PS

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The inverse of the Pascal’s matrix.

Solution Summary: The author explains the procedure for finding the inverse of a square matrix A.

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To find: The inverse of the Pascal’s matrix.

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Chapter 16 Solutions

To find:

The inverse of the Pascal’s matrix.