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2x2 matrix `A= ((3,0),(0,3))` What is `A^2010` ?

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uoor | Student, College Freshman | eNoter

Posted June 10, 2012 at 4:39 PM via web

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2x2 matrix `A= ((3,0),(0,3))`

What is `A^2010` ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 10, 2012 at 5:09 PM (Answer #1)

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You need to evaluate `A^2`  to check if a pattern comes up and this could be used later on, such that:

`A^2 = A*A`

`A^2 = ((3,0),(0,3))*((3,0),(0,3))`

`A^2 = ((3*3+0*0 , 3*0+0*3),(0*3+3*0 , 0*0 + 3*3))`

`A^2 = ((9 , 0),(0, 9))` 

Notice that you may write `A = 3*((1,0),(0,1))`  and `A^2 = 9*((1,0),(0,1)) = 3^2*((1,0),(0,1)).`

You should remember that `((1,0),(0,1)) = I_2` , hence, you may evaluate A^2010 such that:

`A = 3*I_2`

`A^2 = 3^2*I_2`

`A^2010= 3^2010*I_2`

Hence, evaluating `A^2010 ` under the given conditions yields `A^2010= 3^2010*I_2.`

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