When Does (A+B)(A-B) = A^2 - B^2?

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In summary, the distributive property states that when multiplying a sum by a number, each addend can be multiplied separately and then the products can be added together. When applying the distributive property to (A+B)(A-B), the expression can be simplified to A^2 - B^2. It can also be used in reverse to expand and simplify expressions. There are no exceptions to the distributive property and it is closely related to factoring, as it is often used to factor expressions.
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If A and B are n × n matrices, when does
(A + B)(A − B) = A^2 − B^2?

I guess I start out by expanding?

A^2 - BA-B^2+BA=0

Thus AB=BA in order for that to work?
 
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  • #2
charlies1902 said:
If A and B are n × n matrices, when does
(A + B)(A − B) = A^2 − B^2?

I guess I start out by expanding?

A^2 - BA-B^2+BA=0

Thus AB=BA in order for that to work?

Correct.
 

1. What is the distributive property?

The distributive property states that when multiplying a sum by a number, each addend can be multiplied separately and then the products can be added together. In other words, (a + b)c = ac + bc.

2. How does the distributive property apply to (A+B)(A-B)?

When using the distributive property with (A+B)(A-B), the expression can be simplified to A(A-B) + B(A-B). This results in the two terms A^2 - AB and AB - B^2, which can then be combined to give the final answer of A^2 - B^2.

3. Can the distributive property be used in reverse to simplify expressions?

Yes, the distributive property can also be used in reverse to expand and simplify expressions. For example, A(A-B) + B(A-B) can be simplified to (A+B)(A-B).

4. Are there any exceptions to the distributive property?

No, the distributive property holds true for all numbers and variables, including negative numbers and fractions. It is a fundamental property of algebra and is used in many mathematical operations.

5. How is the distributive property related to factoring?

The distributive property is closely related to factoring, as it is often used to factor expressions. For example, in the expression A^2 - B^2, the distributive property can be used to factor out the common factor (A-B), resulting in (A-B)(A+B). This is known as the difference of squares factoring pattern.

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