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Matrices

Question
CBSEENMA12034300
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. space If space straight A comma space straight B space are space square space matrices space of space the space same space order space such space that space AB space equals space BA comma space then space prove space by space induction space that space AB to the power of straight n equals space straight B space to the power of straight n space straight A. space Further comma space prove space that space left parenthesis AB to the power of right parenthesis straight n end exponent space equals space A to the power of n space B to the power of n space for space all space straight n space element of space straight N.

Solution
Here space AB space equals space BA space space We space have space to space prove space that space AB space to the power of straight n equals space straight B space to the power of straight n straight A space space space space space space space space.. left parenthesis 1 right parenthesis space space For space straight n space equals space 1 comma space space straight L. straight H. straight S. space equals space AB space space straight R. straight H. straight S. space equals space BA space space But space AB space equals space BA space left parenthesis given right parenthesis space therefore straight L. straight H. straight S. space equals space straight R. straight H. straight S. space therefore result space left parenthesis 1 right parenthesis space is space true space for space straight n space equals space 1. space Assume space that space the space result space left parenthesis 1 right parenthesis space is space true space for space straight n space equals space straight m. space space therefore left parenthesis AB right parenthesis to the power of straight m space equals space straight A to the power of straight m space straight B to the power of straight m space space Now space AB to the power of straight m plus 1 end exponent space equals space left parenthesis AB right parenthesis to the power of straight m space left parenthesis AB right parenthesis space equals space left parenthesis straight A to the power of straight m space straight B to the power of straight m right parenthesis space left parenthesis AB right parenthesis space space equals space straight A to the power of straight m left parenthesis straight B space to the power of straight m straight A right parenthesis space straight B space equals space straight A to the power of straight m left parenthesis straight A space straight B to the power of straight m right parenthesis space straight B space space space space space space space space space space space space left square bracket because space of left parenthesis 1 right parenthesis right square bracket space space equals space left parenthesis A to the power of m A right parenthesis space left parenthesis B to the power of m space straight B right parenthesis space equals straight A space straight m plus space 1 space straight B space straight m space plus space 1 space space therefore space result space left parenthesis 1 right parenthesis space is space true space for space straight n space equals space straight m space plus space 1 space therefore if space the space result space left parenthesis 1 right parenthesis space is space true space for space straight n space equals space straight m comma space then space it space is space also space true space for space straight n space equals space straight m space plus space 1. space But space the space result space left parenthesis 1 right parenthesis space is space true space for space straight n space equals space 1. space space therefore space by space method space of space induction comma space result space left parenthesis 1 right parenthesis space is space true space for space all space straight n space element of space straight N. space space Hence space the space result.

Some More Questions From Matrices Chapter

If a matrix has 8 elements, what arc the possible orders it can have ?

 

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

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