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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
1 views

For the matrices in Exercise 12, evaluate ( A + B ) 2 and A 2 + 2 A B + B 2 and compare the results.

To determine

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

Explanation

Given information:

Positive integral powers of a square matrix are defined by A1=A and An+1=AnA for every positive integer n.

A=[1240] and B=[3121]

Formula used:

i) Definition: Matrix addition

Addition in Mm×n() is defined by

[aij]m×n+[bij]m×n=[cij]m×n where cij=aij+bij.

ii) Definition: Matrix multiplication

The product of m×n matrix A over and n×p matrix B over is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

columnjofBcolumnjofCrowiofA[ai1ai2ai3ain][b1jb2jb3jbnj]=[cij]rowiofC

Where cij=ai1b1j+ai2b2j++ainbnj.

Explanation:

Let A=[1240] and B=[3121]

Therefore, by using matrix addition

A+B=[1240]+[3121]=[1+3214+20+1]

(A+B)=[4161]

As An+1=AnA for every positive integer n.

Therefore,

(A+B)2=(A+B)(A+B)=[4161][4161]

By using definition of matrix multiplication,

=[(4)(4)+(1)(6)(4)(1)+(1)(1)(6)(4)+(1)(6)(6)(1)+(1)(1)]

=[16+64+124+66+1]

(A+B)2=[225307] ……… (1)

Now,

As An+1=AnA for every positive integer n

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