Explanation Given: The matrix A is, A = [ 1 − 2 2 1 1 − 3 1 − 1 4 ] . The numbers are, 23 13 − 34 31 − 34 63 25 − 17 61 24 14 − 37 41 − 17 − 8 20 − 29 40 38 − 56 116 13 − 11 1 22 − 3 − 6 41 − 53 85 28 − 32 16 . Calculation: The matrix A is, A = [ 1 − 2 2 1 1 − 3 1 − 1 4 ] . The determinant of the matrix A is, | 1 − 2 2 1 1 − 3 1 − 1 4 | = 1 ( 4 − 3 ) + 2 ( 4 + 3 ) + 2 ( − 1 − 1 ) | A | = 1 + 14 − 4 = 11 The inverse of A is A d j A | A | . A d j A is the transpose of the cofactor matrix. Thus the A d j A is, A − 1 = [ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 ] T | A | ... ( 1 ) Find the elements of the above matrix as, c 11 = ( − 1 ) 1 + 1 | 1 − 3 − 1 4 | = 4 − 3 = 1 , c 12 = ( − 1 ) 1 + 2 | 1 − 3 1 4 | = − 1 ( 4 + 3 ) = − 7 , c 13 = ( − 1 ) 1 + 3 | 1 1 1 − 1 | = − 1 − 1 = − 2 c 21 = ( − 1 ) 2 + 1 | − 2 2 − 1 4 | = − 1 ( − 8 + 2 ) = 6 , c 22 = ( − 1 ) 2 + 2 | 1 2 1 4 | = 4 − 2 = 2 , c 23 = ( − 1 ) 2 + 3 | 1 − 2 1 − 1 | = − 1 ( − 1 + 2 ) = − 1 , c 31 = ( − 1 ) 3 + 1 | − 2 2 1 − 3 | = 6 − 2 = 4 , c 32 = ( − 1 ) 3 + 2 | 1 2 1 − 3 | = − 1 ( − 3 − 2 ) = 5 , c 33 = ( − 1 ) 3 + 3 | 1 − 2 1 1 | = 1 + 2 = 3 Thus the cofactor matrix is, [ 1 − 7 − 2 − 6 2 − 1 4 5 3 ] . Write transpose of the above matrix by interchanging the row elements into column elements. [ 1 − 6 4 − 7 2 5 − 2 − 1 3 ] . The equation ( 1 ) becomes, A − 1 = [ 1 11 6 11 4 11 − 7 11 2 11 5 11 − 2 11 − 1 11 3 11 ] . Make the different sets of ( 3 × 1 ) matrix as, [ 23 13 − 34 ] , [ 31 − 34 63 ] , [ 25 − 17 61 ] , [ 24 14 − 37 ] , [ 41 − 17 − 8 ] , [ 20 − 29 40 ] , [ 38 − 56 116 ] , [ 13 − 11 1 ] , [ 22 − 3 − 6 ] , [ 41 − 53 85 ] , [ 28 − 32 16 ] . Write the matrix multiplication of each matrix. [ 23 13 − 34 ] ⋅ A − 1 = [ 23 13 − 34 ] [ 1 11 6 11 4 11 − 7 11 2 11 5 11 − 2 11 − 1 11 3 11 ] = [ 0 18 5 ]

10th Edition

ISBN: 9781337271073

Author: Ron Larson

Publisher: Cengage Learning

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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements ar…

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Chapter 8.PS, Problem 17PS

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The decode form of cryptogram.

Solution Summary: The author explains that the decoded form of cryptogram is REMEMBER, september, and the determinant of the matrix A.

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The decode form of cryptogram.