Proof by Induction: Showing AnAm = An+m

In summary, the conversation involves discussing how to prove a statement involving matrices using induction. The approach is to use one variable as the base case and leave the other variable with a universal quantifier. Alternatively, one can also compute the product of the matrices instead of using induction.
  • #1
annoymage
362
0

Homework Statement



A= (1-n -n

n 1+n)

this is 2x2 matrix, sorry but i don't know which latex to use.


Show that AnAm = An+m

Homework Equations



n/a

The Attempt at a Solution



how do you proof by induction when there's 2 variable?
 
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  • #2
1-n , -n



n , 1+n
 
  • #3
annoymage said:
Show that AnAm = An+m

Homework Equations



n/a

The Attempt at a Solution



how do you proof by induction when there's 2 variable?

You do induction on one of the variables and leave the other with universal
quantifier

You take as base case: For all n [itex] A_n A_0 = A_{0+n} [/itex]

and the induction hypthesis: For all n [itex] A_n A_m = A_{m+n} [/itex]

and try to prove For all n [itex] A_n A_{m+1} = A_{m+1+n} [/itex]
 
  • #4
oooo, i see i see,
thank you very much. :)
 
  • #5
It's actually easier to just compute the product of A_m and A_n
 
  • #6
yeaaa, it is, why didn't i think of that. =.=

hoho thanks again
 

FAQ: Proof by Induction: Showing AnAm = An+m

What is proof by induction?

Proof by induction is a method of mathematical proof used to prove statements that are true for an infinite set of numbers or elements. It involves proving the statement for a base case, and then showing that if the statement is true for any given case, it must also be true for the next case.

Why is proof by induction used?

Proof by induction is used because it allows for the efficient and concise proof of statements that hold true for an infinite set of numbers or elements. It is especially useful in proving properties of sequences, series, and other mathematical structures.

What are the steps involved in a proof by induction?

The steps involved in a proof by induction are as follows: 1) Prove the statement for a base case (usually n = 1 or n = 0). 2) Assume the statement is true for a general case, n = k. 3) Use this assumption to prove that the statement is also true for the next case, n = k + 1. 4) Conclude that the statement is true for all values of n after the base case.

How do you know when to use proof by induction?

Proof by induction is typically used when the statement to be proven involves a recursive or iterative definition, or when the statement applies to all natural numbers (positive integers) or all elements in an infinite set. It can also be used when the statement involves proving the validity of a formula or algorithm.

Are there any limitations to proof by induction?

Proof by induction is limited to statements that can be proven true for all natural numbers or elements in an infinite set. It cannot be used to prove statements that are only true for a finite set of numbers or elements. Additionally, it may not be the most efficient method of proof for certain types of statements.

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