Explanation Given information: Positive integral powers of a square matrix are defined by A 1 = A and A n + 1 = A n ⋅ A for every positive integer n . A = [ 1 2 4 0 ] and B = [ 3 − 1 2 1 ] Formula used: i) Definition: Matrix addition Addition in M m × n ( ℝ ) is defined by [ a i j ] m × n + [ b i j ] m × n = [ c i j ] m × n where c i j = a i j + b i j . ii) Definition: Matrix multiplication The product of m × n matrix A over ℝ and n × p matrix B over ℝ is m × p matrix C = A B , where the element c i j in row i and column j of A B is found by using the elements in row i of A , and the elements in column j of B in the following manner: c o l u m n j o f B c o l u m n j o f C r o w i o f A [ ⋮ ⋮ ⋮ ⋮ a i 1 a i 2 a i 3 ⋯ a i n ⋮ ⋮ ⋮ ⋮ ] ⋅ [ ⋯ b 1 j ⋯ ⋯ b 2 j ⋯ ⋯ b 3 j ⋯ ⋮ ⋯ b n j ⋯ ] = [ ⋮ ⋯ c i j ⋯ ⋮ ] r o w i o f C Where c i j = a i 1 b 1 j + a i 2 b 2 j + ⋯ + a i n b n j . Explanation: Let A = [ 1 2 4 0 ] and B = [ 3 − 1 2 1 ] Therefore, by using matrix addition A + B = [ 1 2 4 0 ] + [ 3 − 1 2 1 ] = [ 1 + 3 2 − 1 4 + 2 0 + 1 ] ( A + B ) = [ 4 1 6 1 ] As A n + 1 = A n ⋅ A for every positive integer n . Therefore, ( A + B ) 2 = ( A + B ) ( A + B ) = [ 4 1 6 1 ] [ 4 1 6 1 ] By using definition of matrix multiplication, = [ ( 4 ) ( 4 ) + ( 1 ) ( 6 ) ( 4 ) ( 1 ) + ( 1 ) ( 1 ) ( 6 ) ( 4 ) + ( 1 ) ( 6 ) ( 6 ) ( 1 ) + ( 1 ) ( 1 ) ] = [ 16 + 6 4 + 1 24 + 6 6 + 1 ] ( A + B ) 2 = [ 22 5 30 7 ] ……… (1) Now, As A n + 1 = A n ⋅ A for every positive integer n

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 1.6, Problem 13E

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To calculate: (A+B)2 and A2+2AB+B2 and compare the results.

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Chapter 1.6, Problem 12E

Chapter 1.6, Problem 14E

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( A + B ) 2 and A 2 + 2 A B + B 2 and compare the results.

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