If $straight A equals open square brackets table row 1 cell space space 2 end cell row 3 cell space space 4 end cell row 5 cell space space 6 end cell end table close square brackets space and space straight B equals open square brackets table row 4 cell space 5 end cell cell space space 6 end cell row 7 cell space 8 end cell cell space space 9 end cell end table close square brackets$ then find AB and BA. Is AB= BA ? What conclusion do you draw?

Solution

straight A equals space open square brackets table row 1 cell space 2 end cell row 3 cell space space 4 end cell row 5 cell space space 6 end cell end table close square brackets comma space straight B equals open square brackets table row 4 cell space 5 space end cell 6 row 7 cell space 8 end cell cell space 9 end cell end table close square brackets

Since number of columns of A = number of rows of B.

therefore space AB space is space definded.

AB equals open square brackets table row 1 cell space 2 end cell row 3 cell space space 4 end cell row 5 cell space space 6 end cell end table close square brackets open square brackets table row 4 cell space space 5 end cell cell space space 6 end cell row 7 cell space space 8 end cell cell space 9 end cell end table close square brackets equals open square brackets table row cell 4 plus 14 end cell cell space 5 plus 16 end cell cell space 6 plus 18 end cell row cell 12 plus 28 end cell cell space space 15 plus 32 end cell cell space 18 plus 36 end cell row cell 20 plus 42 end cell cell space space 25 plus 48 end cell cell space space 30 plus 54 end cell end table close square brackets equals open square brackets table row 18 cell space space 21 end cell cell space space 24 end cell row 40 cell space 47 end cell cell space 54 end cell row 62 cell space 73 end cell cell space space 84 end cell end table close square brackets

Again number of columns of B = number of rows of A

therefore space BA space is space defined.

BA equals space open square brackets table row 4 cell space space 5 end cell cell space space 6 end cell row 7 cell space space 8 end cell cell space space 9 end cell end table close square brackets open square brackets table row 1 cell space space 2 end cell row 3 cell space space 4 end cell row 5 cell space space 6 end cell end table close square brackets equals open square brackets table row cell 4 plus 15 plus 30 end cell cell space space space 8 plus 20 plus 36 end cell row cell 7 plus 24 plus 45 space end cell cell space 14 plus 32 plus 54 end cell end table close square brackets

But space AB not identical to BA space as space AB space and space BA space are space of space different space orders.

We conclude that multiplication of matrices is not commutative.

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