Explanation Given Information: The matrix A = [ 5 7 2 3 ] . Formula used: If A is a n × n matrix, then if there exists an n × n matrix A − 1 such that: A A − 1 = I n and A − 1 A = I n Then A − 1 is the multiplicative inverse of A . For any two matrices, A = [ a b c d ] and B = [ e f g h ] , the product of the two matrices is defined as, A B = [ a e + b g a f + b h c e + d g c f + d h ] Calculation: Consider the matrix: A = [ 5 7 2 3 ] Represent the multiplicative inverse of this matrix A by A − 1 as: A − 1 = [ w x y z ] As the matrix A is a 2 × 2 matrix, the identity matrix will also be a 2 × 2 matrix and denoted by I 2 = [ 1 0 0 1 ] . Use the equation A A − 1 = I 2 to determine the values of w , x , y and z . So, write the equation A A − 1 = I 2 in terms of matrices as: [ 5 7 2 3 ] [ w x y z ] = [ 1 0 0 1 ] Use the formula A B = [ a e + b g a f + b h c e + d g c f + d h ] to multiply the matrices. So, [ 5 ( w ) + 7 ( y ) 5 ( x ) + 7 ( z ) 2 ( w ) + 3 ( y ) 2 ( x ) + 3 ( z ) ] = [ 1 0 0 1 ] Simplify the elements. [ 5 w + 7 y 5 x + 7 z 2 w + 3 y 2 x + 3 z ] = [ 1 0 0 1 ] From this equation, the two systems of linear equations are obtained as: 5 w + 7 y = 1 2 w + 3 y = 0 and 5 x + 7 z = 0 2 x + 3 z = 1 Solve each of these systems by addition method

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ISBN: 8220103679510

Author: Blitzer

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Chapter 8.4, Problem 2CP

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To calculate: The multiplicative inverse of the matrix A=[5723].

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Chapter 8 Solutions

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Ch. 8.1 - Check Point 1 Write the augmented matrix for the...Ch. 8.1 - Check Point 2 Use the matrix ...Ch. 8.1 - Check Point 3 Use matrices to solve the system:...Ch. 8.1 - Check Point 4 Use matrices to solve the system:...Ch. 8.1 - Check Point 5 Solve the system in Check Point. 3...Ch. 8.1 - Prob. 1CVC Ch. 8.1 - Fill in each blank so that the resulting statement...Ch. 8.1 - Prob. 3CVC Ch. 8.1 - Prob. 4CVC Ch. 8.1 - Fill in each blank so that the resulting statement...

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The multiplicative inverse of the matrix A = [ 5 7 2 3 ] .

Solution Summary: The author calculates the multiplicative inverse of the matrix A by A-1.

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To calculate: The multiplicative inverse of the matrix A=[5723].

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