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Asked by majethiyarishat9566.12sdatl | 4th May, 2020, 05:26: PM

Expert Answer:

Question should be:
If A = [2 0 -1 // 5 1 0 // 0 1 3], prove that A-1=A2-6A+11I
Solution:
straight A equals open square brackets table row 2 0 cell negative 1 end cell row 5 1 0 row 0 1 3 end table close square brackets
open vertical bar straight A close vertical bar equals 2 open parentheses 3 minus 0 close parentheses minus 0 open parentheses 15 minus 0 close parentheses minus 1 open parentheses 5 minus 0 close parentheses equals 1
Therefore comma space straight A to the power of negative 1 end exponent space exist
Let apostrophe straight s space find space the space inverse space using space adjoint space of space straight A
straight M subscript 11 equals 3 comma space straight M subscript 12 equals negative 15 comma space straight M subscript 13 equals 5 comma space straight M subscript 21 equals negative 1 comma space straight M subscript 22 equals 6 comma space straight M subscript 23 equals negative 2 comma space straight M subscript 31 equals 1 comma space straight M subscript 32 equals negative 5 comma space straight M subscript 33 equals 2
Adj left parenthesis straight A right parenthesis equals open square brackets table row 3 cell negative 1 end cell 1 row cell negative 15 end cell 6 cell negative 5 end cell row 5 cell negative 2 end cell 2 end table close square brackets
straight A to the power of negative 1 end exponent equals fraction numerator 1 over denominator open vertical bar straight A close vertical bar end fraction Adj left parenthesis straight A right parenthesis equals open square brackets table row 3 cell negative 1 end cell 1 row cell negative 15 end cell 6 cell negative 5 end cell row 5 cell negative 2 end cell 2 end table close square brackets
straight A squared equals open square brackets table row 2 0 cell negative 1 end cell row 5 1 0 row 0 1 3 end table close square brackets open square brackets table row 2 0 cell negative 1 end cell row 5 1 0 row 0 1 3 end table close square brackets equals open square brackets table row 4 cell negative 1 end cell cell negative 5 end cell row 15 1 cell negative 5 end cell row 5 4 9 end table close square brackets
straight A squared minus 6 straight A plus 11 straight I equals open square brackets table row 4 cell negative 1 end cell cell negative 5 end cell row 15 1 cell negative 5 end cell row 5 4 9 end table close square brackets minus open square brackets table row 12 0 cell negative 6 end cell row 30 6 0 row 0 6 18 end table close square brackets plus open square brackets table row 11 0 0 row 0 11 0 row 0 0 11 end table close square brackets
equals open square brackets table row cell 4 minus 12 plus 11 end cell cell negative 1 plus 0 plus 0 end cell cell negative 5 plus 6 plus 0 end cell row cell 15 minus 30 plus 0 end cell cell 1 minus 6 plus 11 end cell cell negative 5 plus 0 plus 0 end cell row cell 5 minus 0 plus 0 end cell cell 4 minus 6 plus 0 end cell cell 9 minus 18 plus 11 end cell end table close square brackets
equals open square brackets table row 3 cell negative 1 end cell 1 row cell negative 15 end cell 6 cell negative 5 end cell row 5 cell negative 2 end cell 2 end table close square brackets
Hence comma space straight A to the power of negative 1 end exponent equals straight A squared minus 6 straight A plus 11 straight I

Answered by Renu Varma | 5th May, 2020, 11:22: AM