Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n times n matrices A and B. a) (a+b)^2=a^2+2ab+b^2 b) (a+b)(a-b)=a^2-b^2

Yulia 2020-12-02 Answered
Explain why each of the following algebraic rules will not work in general when the real numbers a and b are replaced by n×n matrices A and B. a)(a+b)2=a2+2ab+b2
b)(a+b)(ab)=a2b2
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wornoutwomanC
Answered 2020-12-03 Author has 81 answers

Step 1
(a) Consider the matrices A and B of order n.
Here,
(A+B)2=(A+B)(A+B)
(A+B)2=A(A+B)+B(A+B)
(A+B)2=AA+AB+BA+BB
(A+B)2=A2+AB+BA+B2
As n×n matrices are not always commutative, the formula (A+B)2=A2+2AB+B2 will not always work in general for all matrices.
Step 2
(b) Consider the matrices A and B of order n.
Here,
(A+B)(AB)=A(A+B)B(A+B)
(A+B)(AB)=AA+ABBABB
(A+B)(AB)=A2+ABBAB2
As n×n matrices are not always commutative, the formula (A+B)(AB)=A2B2 will not always work in general for all matrices.

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Jeffrey Jordon
Answered 2022-01-22 Author has 2047 answers

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