$If space space straight A equals open square brackets table row cell 1 space space end cell cell space space 0 end cell cell space space space space space 1 end cell row 0 cell space space 1 end cell cell space space space space space 2 end cell row 0 cell space space 0 end cell cell space space space space 4 end cell end table close square brackets$ then show that | 3A | = 27 | A |.

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Solution

straight A equals open square brackets table row cell 1 space space end cell cell space space 0 end cell cell space space space space space 1 end cell row 0 cell space space 1 end cell cell space space space space space 2 end cell row 0 cell space space 0 end cell cell space space space space 4 end cell end table close square brackets therefore space space space space open vertical bar space straight A space close vertical bar space equals space open vertical bar table row 1 cell space space 0 end cell cell space space 1 end cell row 0 cell space space 1 end cell cell space space 2 end cell row 0 cell space space space 0 end cell cell space space 4 end cell end table close vertical bar equals 1 open vertical bar table row 1 cell space space 2 end cell row 0 cell space space 4 end cell end table close vertical bar minus 0 space space open vertical bar table row 0 cell space space 2 end cell row 0 cell space space 4 end cell end table close vertical bar space plus 0 space open vertical bar table row 0 cell space space 1 end cell row 0 cell space space 0 end cell end table close vertical bar space space space space space space space space space space space space space space space equals space 1 space left parenthesis 4 minus 0 right parenthesis minus 0 plus 0 equals 4 Now space space space space 3 straight A equals space open square brackets table row cell 3 space space end cell cell space space 0 end cell cell space space space space space 3 end cell row 0 cell space space 3 end cell cell space space space space space 6 end cell row 0 cell space space 0 end cell cell space space space space 12 end cell end table close square brackets therefore space space space space open vertical bar space 3 straight A space close vertical bar equals space space open square brackets table row 3 cell space space space space 0 end cell cell space space space space space space 3 end cell row 0 cell space space space 3 end cell cell space space space space space 6 end cell row 0 cell space space space 0 end cell cell space space space 12 end cell end table close square brackets space equals 3 space open vertical bar table row 3 cell space space 6 end cell row 0 cell space space 12 end cell end table close vertical bar minus 0 space open vertical bar table row 0 cell space space 3 end cell row 0 cell space 12 end cell end table close vertical bar plus 0 space open vertical bar table row 0 cell space space 3 end cell row 3 cell space space space 6 end cell end table close vertical bar space space space space space space space space space space space space space space space space space equals space 3 space left parenthesis 36 minus 0 right parenthesis minus 0 plus 0 equals 108 space equals 27 space space space open vertical bar space straight A space close vertical bar space therefore space space space space open vertical bar space 3 straight A space close vertical bar space equals space 27 space space space open vertical bar space straight A space close vertical bar space space space space space space space space space space space space space space space space space space space space space space space

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